Alexander Yakhot

Large-scale simulation of the human arterial tree.

1 Full‐scale simulations of the virtual physiological human (VPH) will require significant advances in modelling, multiscale mathematics, scientific computing and further advances in medical imaging. Herein, we review some of the main issues that need to be resolved in order to make three‐dimensional (3D) simulations of blood flow in the human arterial tree feasible in the near future. 2 A straightforward approach is computationally prohibitive even on the emerging petaflop supercomputers, so a three‐level hierarchical approach based on vessel size is required, consisting of: (i) a macrovascular network (MaN); (ii) a mesovascular network (MeN); and (iii) a microvascular network (MiN). We present recent simulations of MaN obtained by solving the 3D Navier–Stokes equations on arterial networks with tens of arteries and bifurcations and accounting for the neglected dynamics through proper boundary conditions. 3 A multiscale simulation coupling MaN–MeN–MiN and running on hundreds of thousands of processors on petaflop computers will require no more than a few CPU hours per cardiac cycle within the next 5 years. The rapidly growing capacity of supercomputing centres opens up the possibility of simulation studies of cardiovascular diseases, drug delivery, perfusion in the brain and other pathologies.

Direct numerical simulation of turbulent flow around a wall-mounted cube: spatio-temporal evolution of large-scale vortices

Flow around a wall-mounted cube is an example of a turbulent flow around a three-dimensional bluff body attached to a surface. The main experimentally observed feature of this type of flow is the appearance of complex vortical structures, e.g. a horseshoe vortex originating in front of the body and enveloping it. The current paper is a follow-up to Yakhot, Liu & Nikitin (2006) in which we presented results of direct numerical simulation (DNS) of turbulent flow around a cube. Here, it is shown that unsteadiness of the considered flow is caused by inviscid-viscous interaction between the horseshoe vortex and the narrow band of positive vorticity attached to the surface in front of the cube. Details of the spatio-temporal evolution of large-scale vortical structures, including samples of long-term visualization and turbulence statistics, are presented. For the normal-to-the-wall velocity, in the vicinity of the cubes front face, the results reveal an anomalous probability distribution, namely, a bimodal distribution and one with high kurtosis.

Numerical investigation of a laminar pulsating flow in a rectangular duct

A pulsating laminar flow of a viscous, incompressible liquid in a rectangular duct has been studied. The motion is induced under an imposed pulsating pressure difference. The problem is solved numerically. Different flow regimes are characterized by a non-dimensional parameter based on the frequency (ω) of the imposed pressure gradient oscillations and the width of the duct (h). This, in fact, is the Reynolds number of the problem at hand. The induced velocity has a phase lag (shift) with respect to the imposed pressure oscillations, which varies from zero at very slow oscillations, to 90° at fast oscillations. The influence of the aspect ratio of the rectangular duct and the pulsating pressure gradient frequency on the phase lag, the amplitude of the induced oscillating velocity, and the wall shear were analyzed

A highly accurate numerical solution of a biharmonic equation

A computational solution of a partial differential equation (PDE) involves a discretization procedure by which the continuous equation is replaced by a discrete algebraic equation. The discretization procedure consists of an approximation of the derivatives in the governing PDE by differences of the dependent variables, which are computed only at discrete points (grid or mesh points). The discretization of the continuous problem inevitably introduces an error in computing the derivatives and, as a result, an error in the computational solution. In general, one starts with a given PDE and uses a discretization procedure for developing a finite-difference equation (FDE) that is a linear relation between discrete values of the unknown function computed on grid point. Then, with the aid of a Taylor series expansion about the node at which the derivative is evaluated, the PDE can be rewritten in the following form: PDE = FDE + TE, where the remainder, TE, is

Direct numerical simulation of turbulent flow in elliptical ducts

Direct numerical simulation (DNS) of fully developed turbulent flow in elliptical ducts is performed. The mean cross-stream secondary flows exhibited by two counterrotating vortices which are symmetrical about the major ellipse’s axis are examined. The mean flow characteristics and turbulence statistics are obtained. The variation of the statistical quantities such as the Reynolds stresses and turbulence intensities along the minor axis of the elliptical cross-section are found to be similar to plane channel data. The turbulent statistics along the major axis are found to be inhibited by the secondary flow transferring high-momentum fluid from the duct’s centre towards the wall. The instantaneous velocity fields in the near-wall region reveal structures similar to the ‘streaks’ except in the vicinity of the major axis endpoints where significant reduction of the turbulent activity due to the wall transverse curvature effect is found.

Numerical simulation of turbulent flow in the inlet region of a smooth pipe

An algebraic-Q4 turbulent eddy viscosity model expresses the eddy viscosity as a solution of a quartic (Q4) equation. The model is applied to numerical simulation of developing turbulent flow in the inlet region of a smooth pipe. Predictions of the flow characteristics, such as velocity profiles accross and along a pipe, pressure drop along a pipe are found in good agreement with experimental data.

An analytical model of a two‐phase liquid metal magnetohydrodynamic generator

An analytical model of a two‐phase liquid‐metal magnetohydrodynamic generator is developed. The model is based on the averaging equations governing the two‐phase flow in the channel of a liquid‐metal magnetohydrodynamic generator. The inhomogeneity of the distribution of the gas phase over the channel cross section is taken into account by introducing the correlation coefficients β1, β2, and β3: 〈αu〉 = β1〈α〉〈u〉, 〈σu〉 = β2〈σ〉〈u〉, 〈ρu〉 = β3〈ρ〉〈u〉. Expressions for these coefficients in terms of the correlation coefficient β1 are obtained. The calculated characteristics of the generator were compared with experimental data obtained at the Argonne National Laboratory. The comparison shows that the calculated and measured characteristics are in fairly good (±15%) agreement.

Phase shift ellipses for pulsating flows

A pulsating laminar flow of a viscous, incompressible fluid through a pipe with an orifice has been studied at relatively low Reynolds numbers. The motion is caused by an imposed sinusoidally varying pressure difference, Δp(t). The induced flow rate, Q(t), has a phase shift, φQ, with respect to the imposed pressure oscillations. We have used that a phase (x,y)-plane instantaneous state plot of two phase-shifted trajectories: x(t)=sin(ωt+φx) and y(t)=sin(ωl+φy), is an ellipse. The ellipses of the instantaneous states Q(t) vs Δp(t) during a cycle allow readily computing the phase shift, φQ.

Low-Reynolds number approximation for turbulent eddy viscosity

We develop a model for turbulent eddy viscosity as a solution of a quartic equation. The model is consistent with the renormalization group (RNG)-based low-Reynolds number approximation for turbulent viscosity. The model has been tested for flow over a backward facing step and the predicted results show very good agreement with the experimental data.

An algebraic-Q4 turbulent eddy viscosity model: boundary layer flow over a flat plate and flow in a pipe

An algebraic turbulent eddy viscosity model is proposed based on a length scale model coupled with the turbulent viscosity expression of the renormalization group theory of turbulence. The eddy viscosity is presented as a solution of a quartic equation. The new length scale model is based on boundary layer characteristics (displacement thickness, shape factor). The model is applied to transitional boundary layer flow over a flat plate and to flow in a smooth pipe. Predictions for the laminar-turbulent transition, and integral characteristics, such as the total skin friction coefficient, mean velocity profile across the boundary layer, and the friction coefficient in a pipe, are found to be in good agreement with experimental data.