Mikael Mortensen

• Piterskaya, Anna & Mortensen, Mikael (2025). A study of the Orr–Sommerfeld and induction equations by Galerkin and Petrov–Galerkin spectral methods utilizing Chebyshev polynomials. Journal of Computational and Applied Mathematics. ISSN 0377-0427. 459. doi: 10.1016/j.cam.2024.116374. Fulltekst i vitenarkiv
• Mortensen, Mikael (2024). A faster multipole Legendre-Chebyshev transform. SIAM Journal on Scientific Computing. ISSN 1064-8275. 46(6), s. A3803–A3826. doi: 10.1137/24M1659352. Fulltekst i vitenarkiv Vis sammendrag

  This paper describes a fast algorithm for transforming Legendre coefficients into Chebyshev coefficients, and vice versa. The algorithm is based on the fast multipole method and is similar to the approach described by Alpert and Rokhlin [SIAM J. Sci. Comput., 12 (1991), pp. 158–179]. The main difference is that we utilize a modal Galerkin approach with Chebyshev basis functions instead of a nodal approach with a Lagrange basis. Part of the algorithm is a novel method that facilitates faster spreading of intermediate results through neighboring levels of hierarchical matrices. This enhancement leads to a method that is approximately 20 percent faster to execute, due to fewer floating point operations. We also describe an efficient initialization algorithm that for the Lagrange basis is roughly 5 times faster than the original method for large input arrays. The described method has both a planning and an execution stage that asymptotically require O(N) flops. The algorithm is simple enough that it can be implemented in 100 lines of vectorized Python code. Moreover, its efficiency is such that a single-threaded C implementation can transform 1,000,000 coefficients in approximately 20 milliseconds on a new MacBook Pro M3, representing about 3 times the execution time of a well-planned (single-threaded) type 2 discrete cosine transform from FFTW (www.fftw.org). Planning for these coefficients requires approximately 50 milliseconds.

• Vasanth, Joel; Rabault, Jean; Alcántara-Ávila, Francisco; Mortensen, Mikael & Vinuesa, Ricardo (2024). Multi-agent Reinforcement Learning for the Control of Three-Dimensional Rayleigh–Bénard Convection. Flow Turbulence and Combustion. ISSN 1386-6184. doi: 10.1007/s10494-024-00619-2. Fulltekst i vitenarkiv
• Piterskaya, Anna; Miloch, Wojciech Jacek & Mortensen, Mikael (2023). A global spectral-Galerkin investigation of a Rayleigh-Taylor instability in plasma using an MHD-Boussinesq model. AIP Advances. 13(10). doi: 10.1063/5.0155976. Fulltekst i vitenarkiv Vis sammendrag

  This paper presents a new efficient algorithm based on the spectral-Galerkin numerical approximations complemented by a magnetohydrodynamics–Boussinesq model and a new solver for studying the development of a Rayleigh–Taylor (RT) instability. We use the Shenfun computational framework in the Cartesian coordinates, which gives the spectral order and accuracy for the considered model based on the magnetohydrodynamics equations and the Boussinesq conjecture. The numerical simulations were conducted for each two- and three-dimensional case, both with and without an external static magnetic field. The validity of the numerical results was examined by comparing the calculated squared L2-norm of the density parameter with the linear stability analysis. We also examined the effects of a uniform tangential magnetic field on the onset and growth of an RT instability at different magnetic field strengths. The analysis of the effectiveness of the presented method suggests that it can be modified for further research on two-component plasma.

• Vignon, Colin; Rabault, Jean; Vasanth, Joel; Alcántara-Ávila, Francisco; Mortensen, Mikael & Vinuesa, Ricardo (2023). Effective control of two-dimensional Rayleigh-Bénard convection: Invariant multi-agent reinforcement learning is all you need. Physics of Fluids. ISSN 1070-6631. 35(6). doi: 10.1063/5.0153181. Fulltekst i vitenarkiv Vis sammendrag

  Rayleigh–Bénard convection (RBC) is a recurrent phenomenon in a number of industrial and geoscience flows and a well-studied system from a fundamental fluid-mechanics viewpoint. In the present work, we conduct numerical simulations to apply deep reinforcement learning (DRL) for controlling two-dimensional RBC using sensor-based feedback control. We show that effective RBC control can be obtained by leveraging invariant multi-agent reinforcement learning (MARL), which takes advantage of the locality and translational invariance inherent to RBC flows inside wide channels. MARL applied to RBC allows for an increase in the number of control segments without encountering the curse of dimensionality that would result from a naive increase in the DRL action-size dimension. This is made possible by the MARL ability for re-using the knowledge generated in different parts of the RBC domain. MARL is able to discover an advanced control strategy that destabilizes the spontaneous RBC double-cell pattern, changes the topology of RBC by coalescing adjacent convection cells, and actively controls the resulting coalesced cell to bring it to a new stable configuration. This modified flow configuration results in reduced convective heat transfer, which is beneficial in a number of industrial processes. We additionally draw comparisons with a conventional single-agent reinforcement learning (SARL) setup and report that in the same number of episodes, SARL is not able to learn an effective policy to control the cells. Thus, our work both shows the potential of MARL for controlling large RBC systems and demonstrates the possibility for DRL to discover strategies that move the RBC configuration between different topological configurations, yielding desirable heat-transfer characteristics.

• Mortensen, Mikael (2022). A Generic and Strictly Banded Spectral Petrov-Galerkin Method for Differential Equations with Polynomial Coefficients. SIAM Journal on Scientific Computing. ISSN 1064-8275. 45(1), s. A123–A146. doi: 10.1137/22M1492842. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we describe a generic spectral Petrov–Galerkin method that is sparse and strictly banded for any linear ordinary differential equation with polynomial coefficients. The method applies to all subdivisions of Jacobi polynomials (e.g., Chebyshev and Legendre), utilizes well-known recurrence relations of orthogonal polynomials, and leads to almost exactly the same discretized system of equations as the integration preconditioners [E. A. Coutsias, T. Hagstrom, and D. Torres, Math. Comp., 65 (1996), pp. 611–635] if this method was redesigned to make use of trial functions that satisfy a given problem’s boundary conditions. A link between the new Petrov–Galerkin method and IP is revealed through a new recursion relation for Jacobi polynomials. Because of the strictly banded nature of all coefficient matrices, the new method extends easily and efficiently to multiple dimensions though the use of tensor product methods.

• Friedemann, Christopher; Mortensen, Mikael & Nossen, Jan (2021). Two-phase co-current flow simulations using periodic boundary conditions in horizontal, 4, 10 and 90° inclined eccentric annulus, flow prediction using a modified interFoam solver and comparison with experimental results. International Journal of Heat and Fluid Flow. ISSN 0142-727X. 88, s. 1–20. doi: 10.1016/j.ijheatfluidflow.2020.108754. Fulltekst i vitenarkiv Vis sammendrag

  Two-phase oil and gas flow were simulated in an entirely eccentric annulus and compared with experimental data at horizontal, 4, 10, and 90° inclination. The gas-phase was sulphur hexafluoride and the liquid phase a mixture of Exxsol D60 and Marcol 82 for the inclined cases (5–16), and pure Exxsol D60 for the horizontal cases (1–4). The diameter of the outer and inner cylinders was 0.1 and 0.04 m, respectively, for the inclined domains and 0.1 and 0.05 m for the horizontal domain. The cases studied consist of liquid phase fractions between 0.3 and 0.65 and mixture velocities from 1.2 to 4.25 m/s. The mean pressure gradient is within 33% of the expected experimental behavior for all inclined cases. In contrast, the low-velocity horizontal domains exhibit significant deviation, with a drastic over-prediction of the mean pressure gradient by as much as 200–335% for cases 1 and 2. The two remaining horizontal cases (3 and 4) are within 22% of the expected mean pressure gradient. Cases 13–16 are a replication of cases 5–8 at an increased inclination; the mean pressure gradient is within 6.5% of the expected increase due to the increase in hydrostatic pressure. By comparing cases 1–4 to previous published simulations at a lower eccentricity, we found a decrease of the mean pressure gradient by 30–40%, which is in line with existing literature, although for single-phase flows. The simulated and experimental liquid holdup profiles are in good agreement when comparing the fractional data; wave and slug frequencies match to within 0.5 Hz; however, at closer inspection, it is apparent that there is a decrease in the amount of phase-mixing of the simulations compared to the experiments. When increasing the mesh density from 115 k cells/m to 2 million cells/m, the simulations exhibit significantly more phase mixing, but are still unable to produce conventional slugs. In a simplified case, conventional slugs are observed at grid sizing of 1 × 1 × 1 mm, whereas the cells of the 2 million cells/m mesh are roughly 1.5 × 1.5 × 1.5 mm.

• Landet, Tormod; Mardal, Kent-Andre & Mortensen, Mikael (2020). Slope limiting the velocity field in a discontinuous Galerkin divergence-free two-phase flow solver. Computers & Fluids. ISSN 0045-7930. 196. doi: 10.1016/j.compfluid.2019.104322. Fulltekst i vitenarkiv Vis sammendrag

  Solving the Navier–Stokes equations when the density field contains a large and sharp discontinuity—such as a water/air free surface—is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that does not compromise on the mass-conservation properties. The equations are discretised using a symmetric interior-penalty discontinuous Galerkin finite element method that is divergence-free to machine precision. A slope limiter made specifically for exactly divergence-free (solenoidal) fields is presented and used to illustrate the difficulties in obtaining convectively stable fields that are also exactly solenoidal. The lessons learned from this are applied in constructing a simpler method based on the use of an existing scalar slope limiter applied to each velocity component. We show by numerical examples how both presented slope limiting methods are vastly superior to the naive non-limited method. The methods can solve difficult two-phase problems with high density-ratios and high Reynolds numbers—typical for marine and offshore water/air simulations—in a way that conserves mass and stops unbounded energy growth caused by the Gibbs phenomenon.

• Friedemann, Christopher; Mortensen, Mikael & Nossen, Jan (2020). Two-phase flow simulations at 0 - 4° inclination in an eccentric annulus. International Journal of Heat and Fluid Flow. ISSN 0142-727X. 83. doi: 10.1016/j.ijheatfluidflow.2020.108586. Fulltekst i vitenarkiv Vis sammendrag

  Co-current two-phase simulations of gas-liquid flow with mixture velocities from 1.2 to 4.2 m/s were run in a partially eccentric annulus and compared with entirely eccentric and concentric experimental data collected at the Institute for Energy Technology in Norway. The gas-phase was sulphur hexafluoride (SF6) for all cases, while the liquid-phase was Exxsol D60 for the horizontal cases and a mixture of Exxsol D60 and Marcol 82 for the inclined case. The outer diameter of the annulus was 0.1 m for all cases, while the inner diameter was 0.05 m in the horizontal configuration and 0.04 m for the inclined configuration. The purpose of this paper is to explore the effect of the holdup fraction, mixture velocity, and interior pipe’s position on the pressure gradient and flow regime, in effect a study of the pressure gradient and holdup fraction transients. The comparisons between simulations and experiments indicate that moving the pipe from an entirely eccentric to the partially eccentric configuration has a drastic impact on the pressure gradient. In all 4 cases where the inner pipe was changed from a completely eccentric geometry in the experiments to a less eccentric configuration in the simulations, we notice an increase of 48–303% of the mean pressure gradient. Comparatively, the 4 cases where the pipe was moved from a concentric experimental configuration to a more eccentric configuration in the simulations result in less drastic pressure gradient changes. Two cases were within 22% of the experimental results for mean, maximum, and minimum pressure gradient, while the last two cases exceeded the minimum and mean pressure gradients by 25–250%, respectively. The flow regime is rarely significantly affected by a change in eccentricity; 2 out of the 8 horizontal cases indicate either a transition from wavy flow to slug flow or significantly larger waves. The most prominent and frequent discrepancies identified were altered slug and wave frequencies. The last case, a 4o inclined, partially eccentric simulation was compared to an entirely eccentric experiment and results in a 0.2 Hz increase in wave frequency, up from the experimental 0.56 Hz and a 49% increase in the mean pressure gradient.

• Marholm, Sigvald; Darian, Diako; Mortensen, Mikael; Miloch, Wojciech Jacek & Marchand, Richard (2020). A Novel Method for Circuits of Perfect Electric Conductors in Unstructured Particle-in-Cell Plasma–Object Interaction Simulations. IEEE Transactions on Plasma Science. ISSN 0093-3813. 48(8), s. 2856–2872. doi: 10.1109/TPS.2020.3010561. Fulltekst i vitenarkiv Vis sammendrag

  A novel numerical method has been developed that incorporates electrically conducting objects into particle-in-cell simulations of electrostatic plasma. The method allows multiple objects connected by voltage and current sources in an arbitrary circuit topology. Moreover, by means of an unstructured mesh, the objects can have arbitrary shapes. The electric potential of the objects is solved self-consistently by incorporating charge constraints into the finite element discretization of the Poisson equation. This method has been implemented in a new code, Particles-in-Unstructured-Cells (PUNC), suitable for rapid prototyping. The flexibility of this code has proven convenient to survey various methods, and an issue of reduced convergence rate of today's unstructured plasma-object interaction codes is highlighted. The results for a conducting sphere immersed in the Maxwellian plasma are in good agreement with previous studies.

• Bergersen, Aslak Wigdahl; Mortensen, Mikael & Sendstad, Kristian Valen (2019). The FDA nozzle benchmark: “In theory there is no difference between theory and practice, but in practice there is”. International Journal for Numerical Methods in Biomedical Engineering. ISSN 2040-7939. 35(1). doi: 10.1002/cnm.3150. Fulltekst i vitenarkiv Vis sammendrag

  The utility of flow simulations relies on the robustness of computational fluid dynamics (CFD) solvers and reproducibility of results. The aim of this study was to validate the Oasis CFD solver against in vitro experimental measurements of jet breakdown location from the FDA nozzle benchmark at Reynolds number 3500, which is in the particularly challenging transitional regime. Simulations were performed on meshes consisting of 5, 10, 17, and 28 million (M) tetrahedra, with Δt   =  10−5 seconds. The 5M and 10M simulation jets broke down in reasonable agreement with the experiments. However, the 17M and 28M simulation jets broke down further downstream. But which of our simulations are “correct”? From a theoretical point of view, they are all wrong because the jet should not break down in the absence of disturbances. The geometry is axisymmetric with no geometrical features that can generate angular velocities. A stable flow was supported by linear stability analysis. From a physical point of view, a finite amount of “noise” will always be present in experiments, which lowers transition point. To replicate noise numerically, we prescribed minor random angular velocities (approximately 0.31% ), much smaller than the reported flow asymmetry (approximately 3% ) and model accuracy (approximately 1% ), at the inlet of the 17M simulation, which shifted the jet breakdown location closer to the measurements. Hence, the high‐resolution simulations and “noise” experiment can potentially explain discrepancies in transition between sometimes “sterile” CFD and inherently noisy “ground truth” experiments. Thus, we have shown that numerical simulations can agree with experiments, but for the wrong reasons.

• Mortensen, Mikael; Dalcin, Lisandro & Keyes, David (2019). mpi4py-fft: Parallel Fast Fourier Transforms with MPI for Python. Journal of Open Source Software (JOSS). 4(36). doi: 10.21105/joss.01340. Fulltekst i vitenarkiv Vis sammendrag

  The fast Fourier transform (FFT) is an algorithm that efficiently computes the discrete Fourier transform. Since the dawn of our digital society, the FFT has permeated to the heart of everyday life applications involving audio, image, and video processing. The importance of the FFT extends beyond signal processing into scientific computing because it diagonalizes the Poisson operator, which is ubiquitous in descriptions of electromagnetism, gravitation, acoustic and elastic waves, potential flow in fluids, stress in solids, Hamiltonians of the Schroedinger’s equation for probability distribution of electrons in quantum mechanics, and diffusion of internal energy, chemical species, and momentum. The FFT has been named the most important numerical algorithm of our time by Professor Gilbert Strang (Strang, 1994).

• Darian, Diako; Marholm, Sigvald; Mortensen, Mikael & Miloch, Wojciech Jacek (2019). Theory and simulations of spherical and cylindrical Langmuir probes in non-Maxwellian plasmas. Plasma Physics and Controlled Fusion. ISSN 0741-3335. 61(8). doi: 10.1088/1361-6587/ab27ff. Fulltekst i vitenarkiv Vis sammendrag

  The collected current by spherical and cylindrical Langmuir probes immersed in an unmagnetized and collisionless non-Maxwellian plasma at rest are theoretically studied, and analytical expressions for the currents of attracted and repelled plasma particles are presented. We consider Kappa, Cairns and the generalized Kappa–Cairns distributions as possible models for the velocity field in the plasma. The current–voltage characteristics curves are displayed and discussed. Furthermore, comparisons with the collected currents in Maxwellian plasmas are given. The results of Particle-in-Cell (PIC) simulations of spherical and cylindrical probes in non-Maxwellian plasmas are also presented, and compared with the theoretical expressions. The results for the collected currents by the Langmuir probes obtained by PIC simulations are in good agreement with the corresponding analytical expressions.

• Marholm, Sigvald; Marchand, Richard; Darian, Diako; Miloch, Wojciech Jacek & Mortensen, Mikael (2019). Impact of Miniaturized Fixed-Bias Multineedle Langmuir Probes on CubeSats. IEEE Transactions on Plasma Science. ISSN 0093-3813. 47(8), s. 3658–3666. doi: 10.1109/TPS.2019.2915810. Fulltekst i vitenarkiv Vis sammendrag

  Kinetic simulation results are presented to study the response of multi-Needle Langmuir Probes of the type used on many satellites. Simulations of isolated probes are used to parametrize the current collected as a function of voltage for a set of densities and temperatures of relevance to Earth ionosphere. These simulations also serve to assess the validity of analytic results obtained from the orbit motion limited (OML) theory used in recent studies. Computed probe characteristics are then fitted with empirical scaling laws and used to account for electron current collected by needle probes on a typical triple CubeSat. These fits are then used to determine the impact of the probes and guards on the spacecraft floating potential for a nominal configuration of bias voltages, over the plasma parameters of interest. In order for the probes to work as intended, they must operate at a positive potential with respect to the ambient plasma. However, results show that for the cases considered, the spacecraft floating potential is so low that the probe with the lowest voltage becomes negative. Possible solutions are examined and proposed to ensure that all probes remain at a positive voltage with respect to surrounding plasma.

• Darian, Diako; Miloch, Wojciech Jacek; Mortensen, Mikael; Miyake, Yohei & Usui, Hideyuki (2019). Numerical simulations of a dust grain in a flowing magnetized plasma. Physics of Plasmas. ISSN 1070-664X. 26(043701), s. 1–14. doi: 10.1063/1.5089631. Fulltekst i vitenarkiv Vis sammendrag

  The effect of an external magnetic field on the formation of the wake in the potential distribution behind a dust grain is studied with self-consistent Particle-In-Cell numerical simulations. The collisionless plasma flow is aligned with the magnetic field. It is demonstrated that the topology of the wakefield is significantly affected by the magnetization degree of plasma and by the ion flow speed. The external magnetic field acts to reduce the potential enhancements in the wake and leads to splitting of the wake pattern across the symmetry axis. For high ion flow speeds, a strong magnetization of plasma suppresses the potential enhancements and results in a narrow negative potential line along the symmetry axis, parallel to the ion flow direction, in the wake.

• Kuchta, Miroslav; Mardal, Kent-Andre & Mortensen, Mikael (2019). Preconditioning trace coupled 3d-1d systems using fractional Laplacian. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 35(1), s. 375–393. doi: 10.1002/num.22304. Fulltekst i vitenarkiv
• Kuchta, Miroslav; Mardal, Kent-Andre & Mortensen, Mikael (2019). On the singular Neumann problem in linear elasticity. Numerical Linear Algebra with Applications. ISSN 1070-5325. 26(1). doi: 10.1002/nla.2212. Fulltekst i vitenarkiv Vis sammendrag

  The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a nonsingular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well‐posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation, we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lamé constant are constructed for the pure displacement formulations, whereas a preconditioner that is robust in both Lamé constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. On the basis of this observation, a modification to the conjugate gradient method is proposed, which achieves optimal error convergence of the computed solution.

• Ketcheson, David I.; Mortensen, Mikael; Parsani, Matteo & Schilling, Nathanael (2019). More efficient time integration for Fourier pseudospectral DNS of incompressible turbulence. International Journal for Numerical Methods in Fluids. ISSN 0271-2091. doi: 10.1002/fld.4773. Fulltekst i vitenarkiv Vis sammendrag

  Time integration of Fourier pseudospectral DNS is usually performed using the classical fourth‐order accurate Runge‐Kutta method or other second‐ or third‐order methods, with a fixed step size. We investigate the use of higher‐order Runge‐Kutta pairs and automatic step size control based on local error estimation. We find that the fifth‐order accurate Runge‐Kutta pair of Bogacki and Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2× to 10× for the same accuracy. Numerical tests (including the Taylor‐Green vortex, Rayleigh‐Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh‐Taylor instability) without compromising accuracy.

• Dalcin, Lisandro; Mortensen, Mikael & Keyes, David (2019). Fast parallel multidimensional FFT using advanced MPI. Journal of Parallel and Distributed Computing. ISSN 0743-7315. 128, s. 137–150. doi: 10.1016/j.jpdc.2019.02.006. Fulltekst i vitenarkiv Vis sammendrag

  We present a new method for performing global redistributions of multidimensional arrays essential to parallel fast Fourier (or similar) transforms. Traditional methods use standard all-to-all collective communication of contiguous memory buffers, thus necessarily requiring local data realignment steps intermixed in-between redistribution and transform steps. Instead, our method takes advantage of subarray datatypes and generalized all-to-all scatter/gather from the MPI-2 standard to communicate discontiguous memory buffers, effectively eliminating the need for local data realignments. Despite generalized all-to-all communication of discontiguous data being generally slower, our proposal economizes in local work. For a range of strong and weak scaling tests, we found the overall performance of our method to be on par and often better than well-established libraries like MPI-FFTW, P3DFFT, and 2DECOMP&FFT. We provide compact routines implemented at the highest possible level using the MPI bindings for the C programming language. These routines apply to any global redistribution, over any two directions of a multidimensional array, decomposed on arbitrary Cartesian processor grids (1D slabs, 2D pencils, or even higher-dimensional decompositions). The high level implementation makes the code easy to read, maintain, and eventually extend. Our approach enables for future speedups from optimizations in the internal datatype handling engines within MPI implementations.

• Friedemann, Christopher; Mortensen, Mikael & Nossen, Jan (2019). Gas-liquid slug flow in a horizontal concentric annulus, a comparison of numerical simulations and experimental data. International Journal of Heat and Fluid Flow. ISSN 0142-727X. 78. doi: 10.1016/j.ijheatfluidflow.2019.108437. Fulltekst i vitenarkiv Vis sammendrag

  Multiphase flow simulations were run in OpenFOAM utilizing InterFoam, a volume of fluid type solver. A concentric annulus geometry was constructed and periodic boundary conditions were applied to alleviate the requirement for a longer domain. The simulations were run in 3, 5 and 7m long domains with the annulus dimensioned such that the outer and inner cylinder diameter were 0.1 and 0.05m respectively. The 4 individual mesh designs were constructed such that the coarsest mesh consists of 102k cells/m while the finest mesh was generated with 400k cells/m. Each mesh was significantly refined within 0.005m of both the inner and outer cylinder wall. The simulation data was compared with experimental pressure and holdup data collected at Institute for Energy Technology in Norway. The 3 and 7m domains reproduced slug frequencies to within 9% of the experiment results of 1.43 Hz for all mesh densities. Comparatively, the 5m domain has larger errors with respect to slugging frequency (22–27%). The 5m case performs poorly, probably due to an artificial restraint introduced by the limitation of available liquid which is set as oil = 0.53 for all cases. The αoil restriction combined with the domain length determines the amount of liquid in the system. This interaction of factors means that the domain length is an important parameter when preparing the simulation. The pressure data display a stronger dependence on the mesh quality in comparison to the slug frequency analysis. The 3m domain with a 400k cells/m mesh resulted in a maximum and minimum pressure gradient of 1783.5 and 803.9 Pa/m, compared to the experiment values of 1785 and 822 Pa/m, which are within 3% of the expected results.

• Mortensen, Mikael (2018). Shenfun: High performance spectral Galerkin computing platform. Journal of Open Source Software (JOSS). 3(31), s. 1071–1071. doi: 10.21105/joss.01071. Fulltekst i vitenarkiv
• Miloch, Wojciech Jacek; Jung, Hendrik; Darian, Diako; Greiner, Franko; Mortensen, Mikael & Piel, A. (2018). Dynamic ion shadows behind finite-sized objects in collisionless magnetized plasma flows. New Journal of Physics. 20(7), s. 1–6. doi: 10.1088/1367-2630/aad066. Fulltekst i vitenarkiv Vis sammendrag

  The potential and density wake behind a finite-sized object in a magnetized collisionless plasma flow is studied with self-consistent numerical simulations. With increasing magnetization of the plasma, the standard picture of ion focusing in the wake for plasmas with large electron to ion temperature ratios becomes invalid. A strong magnetic field parallel to the flow direction leads to a chain of ion depletions in the wake and enhanced ion density at their envelopes. This is due to a novel mechanism of a dynamic ion shadow, which is not the geometrical shadow of the finite-sized object. It corresponds to a change in topology of the wake potential. Complex ion trajectories resulting from electrostatic collisions with the object can lead to significant variations in electrical charging of other objects in the wake.

• Miloch, Wojciech Jacek; Darian, Diako & Mortensen, Mikael (2017). Wake potential of a dust particle in magnetised plasmas. Physica Scripta. ISSN 0031-8949. 92(11). doi: 10.1088/1402-4896/aa90a5. Fulltekst i vitenarkiv Vis sammendrag

  The electric potential distribution around a dust particle immersed in a magnetised supersonic plasma flow is studied by numerical simulations. It is shown that with increasing magnetisation of plasma, the peak in the wake potential gets smaller and moves upstream. For strong magnetisation, the trailing peak in the potential distribution vanishes and the potential becomes more isotropic. The results agree qualitatively with the linear response approach. The numerical simulations are carried out with a particle-in-cell code.

• Darian, Diako; Marholm, Sigvald; Paulsson, Joakim John Paul; Miyake, Y.; Usui, H. & Mortensen, Mikael [Vis alle 7 forfattere av denne artikkelen] (2017). Numerical simulations of a sounding rocket in ionospheric plasma: Effects of magnetic field on the wake formation and rocket potential. Journal of Geophysical Research (JGR): Biogeosciences. ISSN 2169-8953. 122(9), s. 9603–9621. doi: 10.1002/2017JA024284. Fulltekst i vitenarkiv Vis sammendrag

  The charging of a sounding rocket in subsonic and supersonic plasma flows with external magnetic field is studied with numerical particle‐in‐cell (PIC) simulations. A weakly magnetized plasma regime is considered that corresponds to the ionospheric F2 layer, with electrons being strongly magnetized, while the magnetization of ions is weak. It is demonstrated that the magnetic field orientation influences the floating potential of the rocket and that with increasing angle between the rocket axis and the magnetic field direction the rocket potential becomes less negative. External magnetic field gives rise to asymmetric wake downstream of the rocket. The simulated wake in the potential and density may extend as far as 30 electron Debye lengths; thus, it is important to account for these plasma perturbations when analyzing in situ measurements. A qualitative agreement between simulation results and the actual measurements with a sounding rocket is also shown. © 2017 American Geophysical Union

• Haga, Per Thomas; Pizzichelli, Giulia; Mortensen, Mikael; Kuchta, Miroslav; Pahlavian, Soroush Heidari & Sinibaldi, Edoardo [Vis alle 8 forfattere av denne artikkelen] (2017). A numerical investigation of intrathecal isobaric drug dispersion within the cervical subarachnoid space. PLOS ONE. 12(3). doi: 10.1371/journal.pone.0173680. Fulltekst i vitenarkiv Vis sammendrag

  Intrathecal drug and gene vector delivery is a procedure to release a solute within the cerebrospinal fluid. This procedure is currently used in clinical practice and shows promise for treatment of several central nervous system pathologies. However, intrathecal delivery protocols and systems are not yet optimized. The aim of this study was to investigate the effects of injection parameters on solute distribution within the cervical subarachnoid space using a numerical platform. We developed a numerical model based on a patient-specific three dimensional geometry of the cervical subarachnoid space with idealized dorsal and ventral nerve roots and denticulate ligament anatomy. We considered the drug as massless particles within the flow field and with similar properties as the CSF, and we analyzed the effects of anatomy, catheter position, angle and injection flow rate on solute distribution within the cerebrospinal fluid by performing a series of numerical simulations. Results were compared quantitatively in terms of drug peak concentration, spread, accumulation rate and appearance instant over 15 seconds following the injection. Results indicated that solute distribution within the cervical spine was altered by all parameters investigated within the time range analyzed following the injection. The presence of spinal cord nerve roots and denticulate ligaments increased drug spread by 60% compared to simulations without these anatomical features. Catheter position and angle were both found to alter spread rate up to 86%, and catheter flow rate altered drug peak concentration up to 78%. The presented numerical platform fills a first gap towards the realization of a tool to parametrically assess and optimize intrathecal drug and gene vector delivery protocols and systems. Further investigation is needed to analyze drug spread over a longer clinically relevant time frame.

• Friedemann, Christopher; Mortensen, Mikael & Nossen, Jan (2017). Multiphase flow simulation in an annulus configuration. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), MekIT’17 - Ninth national conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE). ISSN 9788494731112. Fulltekst i vitenarkiv Vis sammendrag

  On the 20th of April 2010, explosion and fire killed 11 people on the Deepwater Horizon oil rig in the Gulf of Mexico. After two days in flames, the rig sank and left the riser (the pipe going up from the seabed) on the sea floor leaking oil and gas from the well through the gap between the drilling riser and the drill-string (rod holding the drillbit) inside it. The accident is known as the Macondo accident. Immediate concerns were raised regarding the amount of oil being released into the Gulf of Mexico killing fish and seabirds. The size of the oil spill has also played a significant role in the court proceedings where the operator BP at the end accepted to pay a fine of more than 18 bn US dollars. One of many techniques that were applied was based on using simulation software to match the observed frequency of plugs of oil flowing out of the sunken riser. There are, however, serious flaws to this approach. The most important is the lack of reliable models for multiphase flow in annuli, that is, flow of oil and gas in the ring shaped space between the drillstring and the outer pipe. Commercial simulation software packages for flow of oil and gas in pipes lack the ability to correctly predict pressure drop in annulus flow, even when the flow contains only gas or only liquid. When both gas and oil are flowing in the pipe at the same time, which is called multiphase flow, the models are even worse. Well design is thus based on crude formulas with large uncertainty and high error margins. This may lead to inefficient well design and high risk of production problems. ANNULUS will bring a step change in the scientific knowledge in this field and provide data and physical understanding that can be used as a basis for developing more usable and accurate commercial simulation software in the future. Improved models can be used by operators and contribute to significantly safer and more efficient and profitable well operations and production.

• Mortensen, Mikael (2017). Shenfun - automating the spectral Galerkin method. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), MekIT’17 - Ninth national conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE). ISSN 9788494731112. s. 273–298. doi: 10.21105/joss.01071. Fulltekst i vitenarkiv Vis sammendrag

  With the shenfun Python module (github.com/spectralDNS/shenfun) an effort is made towards automating the implementation of the spectral Galerkin method for simple tensor product domains, consisting of (currently) one non-periodic and any number of periodic directions. The user interface to shenfun is intentionally made very similar to FEniCS (fenicsproject.org). Partial Differential Equations are represented through weak variational forms and solved using efficient direct solvers where available. MPI decomposition is achieved through the mpi4py-fft module (bitbucket.org/mpi4py/mpi4py-fft), and all developed solvers may, with no additional effort, be run on supercomputers using thousands of processors. Complete solvers are shown for the linear Poisson and biharmonic problems, as well as the nonlinear and time-dependent Ginzburg-Landau equation.

• Kuchta, Miroslav; Nordaas, Magne; Verschaeve, Joris C. G.; Mortensen, Mikael & Mardal, Kent-Andre (2016). Preconditioners for saddle point systems with trace constraints coupling 2D and 1D domains. SIAM Journal on Scientific Computing. ISSN 1064-8275. 38(6), s. B962–B987. doi: 10.1137/15M1052822. Fulltekst i vitenarkiv
• Mortensen, Mikael & Langtangen, Hans Petter (2016). High performance Python for direct numerical simulations of turbulent flows. Computer Physics Communications. ISSN 0010-4655. 203, s. 53–65. doi: 10.1016/j.cpc.2016.02.005. Fulltekst i vitenarkiv Vis sammendrag

  Direct Numerical Simulations (DNS) of the Navier Stokes equations is an invaluable research tool in fluid dynamics. Still, there are few publicly available research codes and, due to the heavy number crunching implied, available codes are usually written in low-level languages such as C/C++ or Fortran. In this paper we describe a pure scientific Python pseudo-spectral DNS code that nearly matches the performance of C++ for thousands of processors and billions of unknowns. We also describe a version optimized through Cython, that is found to match the speed of C++. The solvers are written from scratch in Python, both the mesh, the MPI domain decomposition, and the temporal integrators. The solvers have been verified and benchmarked on the Shaheen supercomputer at the KAUST supercomputing laboratory, and we are able to show very good scaling up to several thousand cores. A very important part of the implementation is the mesh decomposition (we implement both slab and pencil decompositions) and 3D parallel Fast Fourier Transforms (FFT). The mesh decomposition and FFT routines have been implemented in Python using serial FFT routines (either NumPy, pyFFTW or any other serial FFT module), NumPy array manipulations and with MPI communications handled by MPI for Python (mpi4py). We show how we are able to execute a 3D parallel FFT in Python for a slab mesh decomposition using 4 lines of compact Python code, for which the parallel performance on Shaheen is found to be slightly better than similar routines provided through the FFTW library. For a pencil mesh decomposition 7 lines of code is required to execute a transform.

• Mortensen, Mikael & Valen-Sendstad, Kristian (2015). Oasis: A high-level/high-performance open source Navier–Stokes solver. Computer Physics Communications. ISSN 0010-4655. 188, s. 177–188. Fulltekst i vitenarkiv Vis sammendrag

  Abstract Oasis is a high-level/high-performance finite element Navier–Stokes solver written from scratch in Python using building blocks from the \{FEniCS\} project (fenicsproject.org). The solver is unstructured and targets large-scale applications in complex geometries on massively parallel clusters. Oasis utilizes \{MPI\} and interfaces, through FEniCS, to the linear algebra backend PETSc. Oasis advocates a high-level, programmable user interface through the creation of highly flexible Python modules for new problems. Through the high-level Python interface the user is placed in complete control of every aspect of the solver. A version of the solver, that is using piecewise linear elements for both velocity and pressure, is shown to reproduce very well the classical, spectral, turbulent channel simulations of Moser et al. (1999). The computational speed is strongly dominated by the iterative solvers provided by the linear algebra backend, which is arguably the best performance any similar implicit solver using \{PETSc\} may hope for. Higher order accuracy is also demonstrated and new solvers may be easily added within the same framework. Program summary Program title: Oasis Catalogue identifier: AEUW_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUW_v1_0.html Program obtainable from: \{CPC\} Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: \{GNU\} Lesser \{GPL\} version 3 or any later version No. of lines in distributed program, including test data, etc.: 3491 No. of bytes in distributed program, including test data, etc.: 266924 Distribution format: tar.gz Programming language: Python/C++. Computer: Any single laptop computer or cluster. Operating system: Any (Linux, OSX, Windows). RAM: a few Megabytes to several hundred Gigabytes Classification: 12. External routines: \{FEniCS\} 1.3.0 (www.fenicsproject.org, that in turn depends on a number of external libraries like MPI, PETSc, Epetra, Boost and ParMetis) Nature of problem: Incompressible, Newtonian fluid flow. Solution method: The finite element method. Unusual features: \{FEniCS\} automatically generates and compiles low-level C++ code based on high-level Python code. Running time: The example provided takes a couple of minutes on a single processor.

• Mortensen, Mikael & Valen-Sendstad, Kristian (2015). Oasis: A high-level/high-performance open source Navier-Stokes solver. Computer Physics Communications. ISSN 0010-4655. 188, s. 177–188. doi: 10.1016/j.cpc.2014.10.026. Fulltekst i vitenarkiv
• Darian, Diako & Mortensen, Mikael (2015). A pseudo-spectral study of Kelvin-Helmholtz instability. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), Proceedings of MekIT'15 Eighth National Conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE). ISSN 9788494424496. s. 133–148. Fulltekst i vitenarkiv
• Bø, Joakim; Bergersen, Aslak Wigdahl; Valen-Sendstad, Kristian & Mortensen, Mikael (2015). Implementation, verification and validation of large eddy simulation models in Oasis. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), Proceedings of MekIT'15 Eighth National Conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE). ISSN 9788494424496. s. 99–122. Fulltekst i vitenarkiv
• Mortensen, Mikael; Kuchta, Miroslav; Mardal, Kent-Andre & Verschaeve, Joris C. G. (2015). BEND|P|Y: Python framework for computing bending of complex plate-beam systems. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), Proceedings of MekIT'15 Eighth National Conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE). ISSN 9788494424496. s. 307–320. Fulltekst i vitenarkiv
• Landet, Tormod; Mortensen, Mikael & Grue, John (2015). Towards a continuous and discontinuous Galerkin method for multi-phase flows. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), Proceedings of MekIT'15 Eighth National Conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE). ISSN 9788494424496. s. 275–294. Fulltekst i vitenarkiv Vis sammendrag

  We present an overview of the continuous and discontinuous Galerkin finite element methods (CG-FEM, DG-FEM) for numerical solution of the transport equation and show how the Navier-Stokes equations can be solved with CG-FEM in both a coupled and segregated manner. We show convergence of the implemented Navier-Stokes solvers for single-phase ows on the well known Taylor-Green vortex and also show how the volume of uid (VOF) method for tracking the interface between two uids can be implemented in DG-FEM. Our work is aimed towards creating a method for looking at the impact forces from water waves on structures and comments are made on how to evolve the presented solution methods into a two-phase solver that can handle the physics involved in highly non-linear free surface simulations.

• Kuchta, Miroslav; Mardal, Kent-Andre & Mortensen, Mikael (2015). Characterization of the space of rigid motions in arbitrary domains. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), Proceedings of MekIT'15 Eighth National Conference on Computational Mechanics. International Center for Numerical Methods in Engineering (CIMNE). ISSN 9788494424496. s. 259–274. Fulltekst i vitenarkiv
• Sayed, Ahmad El; Mortensen, Mikael & Wen, John Z. (2014). Assessment of the presumed mapping function approach for the stationary laminar flamelet modelling of reacting double scalar mixing layers. Combustion theory and modelling. ISSN 1364-7830. 18, s. 552–581. doi: 10.1080/13647830.2014.939229. Fulltekst i vitenarkiv
• Mortensen, Mikael; Mardal, Kent-Andre; Martin, B & Pahlavian, S. (2013). Preliminary study of the impact of spinal cord nerve roots and denticulate ligaments on drug movement in the cervical spinal subarachnoid space. I Mardal, Kent-Andre; Logg, Anders Bernhard & Massing, Andre (Red.), Proceedings of the 26th Nordic Seminar on Computational Mechanics. Simula Research Laboratory. ISSN 9788292593127. s. 213–216. Fulltekst i vitenarkiv
• Kuchta, Miroslav & Mortensen, Mikael (2013). A second order fast sweeping method for the Eikonal equation based on minimization. I Mardal, Kent-Andre; Logg, Anders Bernhard & Massing, Andre (Red.), Proceedings of the 26th Nordic Seminar on Computational Mechanics. Simula Research Laboratory. ISSN 9788292593127. s. 145–148. Fulltekst i vitenarkiv
• Valen-Sendstad, Kristian; Logg, Anders; Mardal, Kent-Andre; Narayanan, Harish; Mortensen, Mikael & Wells, Garth (2012). A Comparison of Some Common Finite Element Schemes for the Incompressible Navier-Stokes Equations. I Logg, Anders; Mardal, Kent-Andre & Wells, Garth N. (Red.), Automated Solution of Differential Equations by the Finite Element Method. Springer. ISSN 9783642230981. s. 395–418. Fulltekst i vitenarkiv
• Valen-Senstad, Kristian; Logg, Anders Bernhard; Mardal, Kent-Andre; Narayanan, Harish & Mortensen, Mikael (2012). A comparison of finite element schemes for the incompressible Navier–Stokes equations. I Logg, Anders; Mardal, Kent-Andre & Wells, Garth N. (Red.), Automated Solution of Differential Equations by the Finite Element Method. Springer. ISSN 9783642230981. s. 399–420. Fulltekst i vitenarkiv
• Mortensen, Mikael; Mardal, Kent-Andre & Langtangen, Hans Petter (2012). Simulation of transitional flows. I Logg, Anders; Mardal, Kent-Andre & Wells, Garth N. (Red.), Automated Solution of Differential Equations by the Finite Element Method. Springer. ISSN 9783642230981. s. 421–440. Fulltekst i vitenarkiv
• Valen-Senstad, Kristian; Mardal, Kent-Andre; Mortensen, Mikael; Reif, Bjørn Anders Pettersson & Langtangen, Hans Petter (2011). On The Complexity of Stationary Flow in Patient–Specific Aneurysms. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), MekIT'11: Sixth National Conference on Computational Mechanics, Trondheim 23-24 May 2011. Tapir Akademisk Forlag. ISSN 9788251927987. Fulltekst i vitenarkiv Vis sammendrag

  Summary Sound or bruits in the vasculatory system at the range of 1Hz-1kHz is known to occur, and frequency peaks at high frequencies are associated with sick blood vessels. The exact reason for these sounds are not known, but the sound may be generated by turbulence. Turbulence generated sound is typically associated with high frequencies but also with low energy levels. Unfortunately, most blood flow simulations are performed with space and time resolutions where rapid fluctuations typ- ically associated with non-laminar flow cannot be captured. Complex non-laminar flow and the subsequent oscillatory wall stresses can po- tentially be of importance in understanding aneurysm rupture through the local arterial wall response. To investigate the possible presence of complex non-laminar blood flow in intracranial aneurysms, we per- formed high resolution computational fluid dynamics (CFD) simulations for patient specific middle cerebral artery (MCA) aneurysms with sta- tionary inflow conditions. The current study confirms, through high resolution simulations, that complex oscillatory flow may occur in in- tracranial aneurysms. CFD practitioners should therefore be aware of the potentially strong demands to space-time computational resolution in blood flow simulations.

• Mortensen, Mikael; Langtangen, Hans Petter & Myre, Jørgen (2011). CBC.RANS - a new flexible, programmable software framework for computational fluid dynamics. I Skallerud, Bjørn Helge & Andersson, Helge Ingolf (Red.), MekIT'11: Sixth National Conference on Computational Mechanics, Trondheim 23-24 May 2011. Tapir Akademisk Forlag. ISSN 9788251927987. Fulltekst i vitenarkiv
• Reif, Bjørn Anders Pettersson & Mortensen, Mikael (2011). A Nonlinear Eddy-Viscosity Model for Near-Wall Turbulence. ERCOFTAC Series. ISSN 1382-4309. 14, s. 269–276. doi: 10.1007/978-90-481-9603-6_28. Fulltekst i vitenarkiv
• Mortensen, Mikael; Langtangen, Hans Petter & Wells, Garth N. (2011). A FEniCS-based programming framework for modeling turbulent flow by the Reynolds-averaged Navier-Stokes equations. Advances in Water Resources. ISSN 0309-1708. 34(9), s. 1082–1101. doi: 10.1016/j.advwatres.2011.02.013. Fulltekst i vitenarkiv
• Valen-Senstad, Kristian; Mardal, Kent-Andre; Mortensen, Mikael; Reif, Bjørn Anders Pettersson & Langtangen, Hans Petter (2011). Direct numerical simulation of transitional flow in a patient-specific intracranial aneurysm. Journal of Biomechanics. ISSN 0021-9290. 44(16), s. 2826–2832. doi: 10.1016/j.jbiomech.2011.08.015. Fulltekst i vitenarkiv
• Mortensen, Mikael; Reif, Bjørn Anders Pettersson & Wasberg, Carl Erik (2010). Assessment of the finite volume method applied to the nu(2)-f model. International Journal for Numerical Methods in Fluids. ISSN 0271-2091. 63(4), s. 495–516. doi: 10.1002/fld.2091. Fulltekst i vitenarkiv Vis sammendrag

  The objective of this paper is to assess the accuracy of low-order finite volume (FV) methods applied to the nu(2)-f turbulence model of Durbin (Theoret. Comput. Fluid Dyn. 1991; 3:1-13) in the near vicinity of solid walls. We are not (like many others) concerned with the stability of solvers - the topic at hand is simply whether the mathematical properties of the nu(2)-f model can be captured by the given, widespread, numerical method. The nu(2)-f model is integrated all the way up to solid walls, where steep gradients in turbulence parameters are observed. The full resolution of wall gradients imposes quite high demands on the numerical schemes and it is not evident that common (second order) FV codes can fully cope with such demands. The nu(2)-f model is studied in a statistically one-dimensional, fully developed channel flow where we compare FV schemes with a highly accurate spectral element reference implementation. For the FV method a higher-order face interpolation scheme, using Lagrange interpolation polynomials up to arbitrary order, is described. It is concluded that a regular second-order FV scheme cannot give an accurate representation of all model parameters, independent of mesh density. To match the spectral element solution an extended source treatment (we use three-point Gauss-Lobatto quadrature), as well as a higher-order discretization of diffusion is required. Furthermore, it is found that the location of the first internal node need to be well within y(+) = 1. Copyright (C) 2009 John Wiley & Sons, Ltd.

• Reif, Bjørn Anders Pettersson; Mortensen, Mikael & Langer, Carlos A. (2009). Towards Sensitizing the Nonlinear v(2)-f Model to Turbulence Structures. Flow Turbulence and Combustion. ISSN 1386-6184. 83(2), s. 185–203. doi: 10.1007/s10494-008-9194-7. Fulltekst i vitenarkiv Vis sammendrag

  In this paper a one-way coupling between the nonlinear v (2) -aEuro parts per thousand f model by Pettersson Reif (Flow Turbul Combust 76:241-256, 2006) and an algebraic structure-based model have been investigated. Comparisons with available experimental and numerical data indicate that the compatibility between the two models is good and that their joint performance is satisfactory in the cases considered here. A full coupling between the models seems therefore a potentially viable route towards a significant advancement of engineering turbulence models and their predictive capabilities.

• Mortensen, Mikael & Bilger, Robert W. (2008). Derivation of the conditional moment closure equations for spray combustion. Combustion and Flame. ISSN 0010-2180. 155(3), s. 369–369. doi: 10.1016/j.combustflame.2008.10.013. Fulltekst i vitenarkiv Vis sammendrag

  In this work we derive the fundamental equations for conditional moment closure (CMC) modelling of individual phases set in a two-phase flow. The derivation is based on the instantaneous transport equations for the single phase that involve a level set/indicator function technique for accounting for interfaces. Special emphasis is put on spray combustion with the CMC equations formulated for the gas phase. The CIVIC equations are to be viewed as an adjunct to existing methods for the modelling of the dynamics of sprays: they provide a refinement of the modelling of chemical reactions in the gas phase. The resulting CIVIC equations differ significantly from those already in use in the literature. They contain, of course, unclosed terms that need to be modelled. Investigation of the unclosed terms associated with evaporation at the droplet surface is well beyond the capabilities of laboratory measurement or direct numerical simulation. it is proposed that modelling of these terms be based on the well-established 'laws' of similarity between heat and mass transfer: an example is detailed for one example of the general modelling of the spray dynamics. other unclosed terms are important throughout the gas phase. Models used for these terms in single-phase flows are reviewed and it is proposed that any modifications needed for these models be investigated by DNS of suitable model problems having good resolution of the flow and mixing in the inter-droplet space. It is proposed that a spray analogue of the scalar mixing layer that has been widely studied in single-phase flows be used as the model problem for such DNS studies and also for LES and RANS modelling. (c) 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

• Mortensen, Mikael; Kops, Stephen M de Bruyn & Cha, Chong M. (2007). Direct numerical simulations of the double scalar mixing layer Part II: Reactive scalars. Combustion and Flame. ISSN 0010-2180. 149(4), s. 392–408. doi: 10.1016/j.combustflame.2007.03.001. Fulltekst i vitenarkiv Vis sammendrag

  The reacting double scalar mixing layer (RDSML) is investigated as a canonical multistream flow and a model problem for simple piloted diffusion flames. In piloted diffusion flames, the reacting fuel and oxidizer streams are initially separated by a central pilot stream at stoichiometric composition. The primary purpose of this pilot is to delay the mixing of the pure streams until a stable flame base can develop. In such multistrearn systems, the modeling of turbulent scalar mixing is complicated by the multiple feed streams, leading to more complex fine-scale statistics, which remain as yet an unmet modeling challenge compared to the simpler two-feed system. In Part I we described how multimodal mixture fraction probability density functions (PDFs) and conditional scalar dissipation rates can be modeled with a presumed mapping function approach. In this work we present an efficient and robust extension of the modeling to a general multistrearn reacting flow and compare predictions to three-dimensional direct numerical simulations (DNS) of the RDSML with a single-step reversible chemistry model and varying levels of extinction. With high extinction levels, the interaction with the pilot stream is described. Additionally, state-of-the-art combustion modeling calculations including conditional moment closure (CMC) and stationary laminar flamelet modeling (SLFM) are performed with the newly developed mixing model. Excellent agreement is found between the DNS and modeling predictions, even where the PDF is essentially a triple-delta shape near the flame base, so long as extinction levels are moderate to low. The suggested approach outlined in this paper is strictly valid only for flows that can be described by a single mixture fraction. For these flows the approach should provide engineers with fine-scale models that are of accuracy comparable to those already available for binary mixing, at only marginally higher complexity and cost. (C) 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Se alle arbeider i NVA

• Miloch, Wojciech Jacek; Usui, Hideyuki; Miyake, Yohei; Mortensen, Mikael; Chernyshov, A.A. & Baddeley, Lisa [Vis alle 11 forfattere av denne artikkelen] (2021). Long-term and sustainable research- based student mobility projects within space weather. Fulltekst i vitenarkiv
• Friedemann, Christopher; Mortensen, Mikael & Nossen, Jan (2017). Multiphase flow simulation in an annulus configuration. Fulltekst i vitenarkiv Vis sammendrag

  On the 20th of April 2010, explosion and fire killed 11 people on the Deepwater Horizon oil rig in the Gulf of Mexico. After two days in flames, the rig sank and left the riser (the pipe going up from the seabed) on the sea floor leaking oil and gas from the well through the gap between the drilling riser and the drill-string (rod holding the drillbit) inside it. The accident is known as the Macondo accident. Immediate concerns were raised regarding the amount of oil being released into the Gulf of Mexico killing fish and seabirds. The size of the oil spill has also played a significant role in the court proceedings where the operator BP at the end accepted to pay a fine of more than 18 bn US dollars. One of many techniques that were applied was based on using simulation software to match the observed frequency of plugs of oil flowing out of the sunken riser. There are, however, serious flaws to this approach. The most important is the lack of reliable models for multiphase flow in annuli, that is, flow of oil and gas in the ring shaped space between the drillstring and the outer pipe. Commercial simulation software packages for flow of oil and gas in pipes lack the ability to correctly predict pressure drop in annulus flow, even when the flow contains only gas or only liquid. When both gas and oil are flowing in the pipe at the same time, which is called multiphase flow, the models are even worse. Well design is thus based on crude formulas with large uncertainty and high error margins. This may lead to inefficient well design and high risk of production problems. ANNULUS will bring a step change in the scientific knowledge in this field and provide data and physical understanding that can be used as a basis for developing more usable and accurate commercial simulation software in the future. Improved models can be used by operators and contribute to significantly safer and more efficient and profitable well operations and production.

• Kuchta, Miroslav; Mortensen, Mikael & Mardal, Kent-Andre (2016). Solving singular problems. The right way. Fulltekst i vitenarkiv
• Bergersen, Aslak; Mortensen, Mikael & Valen-Sendstad, Kristian (2016). A Benchmark Standard Model of a Generic Medical Device; All Models Are Wrong, but Some Are Useful. Fulltekst i vitenarkiv
• Bergersen, Aslak; Mortensen, Mikael & Valen-Sendstad, Kristian (2016). The FDA Nozzle Benchmark: In Theory There Is No Difference Between Theory and Practice, but in Practice There Is. Fulltekst i vitenarkiv
• Bergersen, Aslak; Valen-Sendstad, Kristian & Mortensen, Mikael (2016). The Oasis Navier-Stokes Solver: Theory, Accuracy, Applications, and High-Performance Computing Capabilities. Fulltekst i vitenarkiv
• Mortensen, Mikael (2015). Massively parallel implementation in Python of a pseudo-spectral DNS code for turbulent flows. Fulltekst i vitenarkiv Vis sammendrag

  https://www.euroscipy.org/2015/schedule/presentation/6/

• Mortensen, Mikael; Mardal, Kent-Andre; Martin, Bryn & Pahlavian, Soroush (2014). Preliminary study of the impact of spinal cord nerve roots and denticulate ligaments on drug movement in the cervical spinal subarachnoid space. Fulltekst i vitenarkiv
• Mardal, Kent-Andre; Støverud, Karen-Helene; Mortensen, Mikael; Jain, Kartik & Martin, Bryn (2014). Cerebrospinal Fluid Flow in Association With the Central Nervous System - Chiari and Drug Delivery. Fulltekst i vitenarkiv
• Mardal, Kent-Andre; Støverud, Karen-Helene; Mortensen, Mikael; Jain, Kartik; Martin, Bryn & Pahlavian, Soroush (2014). Cerebrospinal fluid flow in association with the central nervous system -- Chiari and drug delivery. Fulltekst i vitenarkiv
• Valen-Sendstad, Kristian; Mardal, Kent-Andre; Mortensen, Mikael; Langtangen, Hans; Reif, Bjørn & Steinman, David (2011). 'Turbulence' in Cerebral Aneurysms. Fulltekst i vitenarkiv
• Valen-Sendstad, Kristian; Mardal, Kent-Andre; Mortensen, Mikael; Reif, Bjørn & Langtangen, Hans (2011). Direct Numerical Simulation of Transitional Flow in Patient-Specific Intracranial Aneurysms. Fulltekst i vitenarkiv
• Valen-Sendstad, Kristian; Mardal, Kent-Andre; Mortensen, Mikael; Reif, Bjørn & Langtangen, Hans (2011). Simulation Methodology for Cerebral Blood Flow. Fulltekst i vitenarkiv Vis sammendrag

  In experiments turbulence has previously been shown to occur in intracranial aneurysms, but the turbulent effects on the arterial wall have not been investigated. The effects of turbulence induced oscillatory wall stresses could be of great importance in understanding aneurysm rupture due to the local arterial wall response to such hemodynamical forces. To investigate the effects of turbulence on blood flow in an intracranial aneurysm, we performed a high resolution computational fluid dynamics (CFD) simulations in a patient specific middle cerebral artery (MCA) aneurysm using a realistic, pulsatile inflow velocity. The flow showed transition to turbulence just after peak systole, and the turbulent fluctuations increased in intensity until mid deceleration, before relaminarization occurred during diastole. The flow impinged with an angle nearly normal onto the aneurysm dome, and the turbulence structures greatly affected both the frequency of change of wall shear stress (WSS) direction and WSS magnitude, which reached a maximum value of 41.5 Pa. The recorded frequencies were predominantly in the range of 1-500 Hz. The current study confirms, through properly resolved CFD simulations, that turbulence can occur in intracranial aneurysms. Because of the local arterial wall response to hemodynamical forces, the effects of oscillatory WSS caused by turbulence, could be correlated with aneurysm rupture, and should be investigated experimentally.

• Valen-Sendstad, Kristian; Mardal, Kent-Andre; Mortensen, Mikael; Reif, Bjørn & Langtangen, Hans (2011). Computational Cerebral Hemodynamics. Fulltekst i vitenarkiv
• Valen-Sendstad, Kristian; Mortensen, Mikael; Mardal, Kent-Andre; Reif, Bjørn & Langtangen, Hans (2011). Investigations of Transistional Flows Using Image Based Hemodynamics. Fulltekst i vitenarkiv Vis sammendrag

  We investigated the complexity of blood flow in intracranial aneurysms in terms of the frequency spectrum of pointwise velocity and pressure fluctuations. The investigations were performed using high-resolution computational fluid dynamics (CFD) simulations for patient-specific middle cerebral artery (MCA) aneurysms, using realistic inflow profile. Through these simulations, the study confirmed that complex transitional flow can occur in intracranial aneurysms. In particular, we found fluctuations of the order of 500- 1000 Hz.

• Valen-Sendstad, Kristian; Mardal, Kent-Andre; Mortensen, Mikael; Reif, Bjørn; Langtangen, Hans & Skallerud, Bjørn [Vis alle 7 forfattere av denne artikkelen] (2011). On the Complexity of Stationary Flow in Patient-Specific Aneurysms. Fulltekst i vitenarkiv Vis sammendrag

  Summary Sound or bruits in the vasculatory system at the range of 1Hz-1kHz is known to occur, and frequency peaks at high frequencies are associated with sick blood vessels. The exact reason for these sounds are not known, but the sound may be generated by turbulence. Turbulence generated sound is typically associated with high frequencies but also with low energy levels. Unfortunately, most blood flow simulations are performed with space and time resolutions where rapid fluctuations typ- ically associated with non-laminar flow cannot be captured. Complex non-laminar flow and the subsequent oscillatory wall stresses can po- tentially be of importance in understanding aneurysm rupture through the local arterial wall response. To investigate the possible presence of complex non-laminar blood flow in intracranial aneurysms, we per- formed high resolution computational fluid dynamics (CFD) simulations for patient specific middle cerebral artery (MCA) aneurysms with sta- tionary inflow conditions. The current study confirms, through high resolution simulations, that complex oscillatory flow may occur in in- tracranial aneurysms. CFD practitioners should therefore be aware of the potentially strong demands to space-time computational resolution in blood flow simulations.

• Mardal, Kent-Andre; Valen-Sendstad, Kristian; Alnæs, Martin; Mortensen, Mikael; Langtangen, Hans & Reif, Bjørn (2011). The Finite Element Library FEniCS and Simulation of Blood Flow in Cerebral Aneurysms. Fulltekst i vitenarkiv
• Kjeldsberg, Henrik Aasen; Valen-Senstad, Kristian; Sundnes, Joakim & Mortensen, Mikael (2025). The Clot Thickens: Investigating Thrombus Formation Through Left Atrial Flow Simulations. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  Atrial fibrillation (AF) is a common cardiac arrhythmia characterized by an irregular and often rapid heartbeat, which significantly increases the risk of thrombus formation and subsequent stroke. While there is no definitive cure for AF, patients are typically managed With medications, such as anticoagulants, to reduce stroke risk. The decision to undergo such treatment is often determined by clinical risk scores, such as the CHA2DS2–VASc score, which assesses stroke risk based on factors like age, hypertension, and prior stroke history. However, these scores do not account for patient-specific cardiac geometry or blood flow patterns. This dissertation addresses this limitation by using high-resolution computational fluid dynamics (CFD) simulations to quantify patient-specific cardiac anatomy and blood flow. We investigate how these factors contribute to thrombus formation and, by extension, stroke, and assess their potential as complementary markers to existing clinical risk scores. In pursuit of this goal, we developed an open-source, moving-domain CFD solver (OasisMove) and an automated modeling pipeline (VaMPy) to enable high-resolution, high-performance simulations of left atrial (LA) hemodynamics in patients with AF. We first verify and validate these tools using benchmark problems and experimental data. We then systematically compare the impact of rigid, generic, and patient-specific wall motion on LA hemodynamics, demonstrating that incorporating patient-specific motion notably alters flow patterns and proposed indicators of thrombus formation. To address the Challenge of prescribing inflow conditions in patient-specific simulations, we introduce an anatomically based and physiologically constrained method for estimating inlet boundary conditions. This approach enables physiologically plausible flow without the need for invasive measurements. Finally, we apply our workflow to a larger cohort of AF patients and show that combining CFD-derived hemodynamic metrics and morphological parameters with traditional clinical scores results in stronger correlations with thrombus presence and stroke history than clinical scores alone. Overall, our framework provides deeper Insight into AF-related flow phenomena, and results from its application to a patient cohort suggests that CFD-derived metrics and morphological parameters could be valuable additions to stroke risk assessment in AF patients.

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