Variational Multiscale Large Eddy Simulation Research Articles

Computational fluid dynamics of blood flow in an idealized left human heart.

We construct an idealized computational model of the left human heart for the study of the blood flow dynamics in the left atrium and ventricle. We solve the Navier-Stokes equations in the ALE formulation and we prescribe the left heart wall displacement based on physiological data; moreover, we consider the presence of both the mitral and aortic valves through the resistive method. We simulate the left heart hemodynamics by means of the finite element method and we consider the variational multiscale large eddy simulation (LES) formulation to account for the transitional and nearly turbulent regimes of the blood flow in physiological conditions. The main contribution of this paper is the characterization of the blood flow in an idealized configuration of the left heart aiming at reproducing function in normal conditions. Our assessment is based on the analysis of instantaneous and phase averaged velocity fields, blood pressure, and other clinically meaningful fluid dynamics indicators. Finally, we show that our idealized computational model can be suitably used to study and critically discuss pathological scenarios like that of a regurgitant mitral valve.

Separation delay through contoured transverse grooves on a 2D boat-tailed bluff body: Effects on drag reduction and wake flow features

Abstract The effectiveness of properly contoured transverse grooves in delaying the flow separation occurring on a two-dimensional boat-tailed bluff body is assessed through numerical simulations. The body has a cross-section with a 3:1 elliptical forebody and a rectangular main part followed by a circular-arc boat tail. Three-dimensional Variational Multiscale Large Eddy Simulations are carried out at R e = D u ∞ ∕ ν = 9 . 6 × 1 0 4 , using a mixed finite-volume/finite-element method. The introduction of one contoured groove on each of the boat-tail lateral surfaces produces a significant delay of flow separation and a consequent increase of the base pressure, with a global drag reduction of the order of 9 . 7 % . The wake dynamical structure remains qualitatively similar to the one typical of blunt-based two-dimensional bodies, with quantitative variations that are consistent with the reduction in wake width caused by boat tailing and by the grooves. The introduction of the grooves leads also to a regularization of the vortex shedding downstream of the body, which is more correlated in the spanwise direction. Finally, a few supplementary simulations show that the effect of the grooves is also robust to the variation of the geometrical parameters defining their location and shape.

Separation control and drag reduction for boat-tailed axisymmetric bodies through contoured transverse grooves

We describe the results of a numerical and experimental investigation aimed at assessing the performance of a control method to delay boundary layer separation consisting of the introduction on the surface of contoured transverse grooves, i.e. of small cavities with an appropriate shape orientated transverse to the incoming flow. The shape of the grooves and their depth – which must be significantly smaller than the thickness of the incoming boundary layer – are chosen so that the flow recirculations present within the grooves are steady and stable. This passive control strategy is applied to an axisymmetric bluff body with various rear boat tails, which are characterized by different degrees of flow separation. Variational multiscale large eddy simulations and wind tunnel tests are carried out. The Reynolds number, for both experiments and simulations, is $Re=u_{\infty }D/\unicode[STIX]{x1D708}=9.6\times 10^{4}$; due to the different incoming flow turbulence level, the boundary layer conditions before the boat tails are fully developed turbulent in the experiments and transitional in the simulations. In all cases, the introduction of one single axisymmetric groove in the lateral surface of the boat tails produces significant delay of the boundary layer separation, with consequent reduction of the pressure drag. Nonetheless, the wake dynamical structure remains qualitatively similar to the one typical of a blunt-based axisymmetric body, with quantitative variations that are consistent with the reduction in wake width caused by boat tailing and by the grooves. A few supplementary simulations show that the effect of the grooves is also robust to the variation of the geometrical parameters defining their shape. All the obtained data support the interpretation that the relaxation of the no-slip boundary condition for the flow surrounding the recirculation regions, with an appreciable velocity along their borders, is the physical mechanism responsible for the effectiveness of the present separation-control method.

Connection between base drag, separating boundary layer characteristics and wake mean recirculation length of an axisymmetric blunt-based body

The variation of the base drag of an axisymmetric bluff body caused by modifications of the boundary-layer separating at the sharp-edged contour of its base is analysed through different numerical simulations, and the results are compared with those of a previous experimental investigation. Variational MultiScale Large-Eddy Simulations (VMS-LES) are first carried out on the same nominal geometry and at the same Reynolds number of the experiments. Subsequently, Direct Numerical Simulations (DNS) are performed at Reynolds numbers that are roughly two orders of magnitude lower, in order to investigate on the sensitivity of the main findings to the Reynolds number. The results of experiments, VMS-LES and DNS simulations show that an increase of the base pressure – and thus a decrease of the base drag – may be obtained by increasing the boundary layer thickness before separation, which causes a proportional increase of the length of the mean recirculation region behind the body. In spite of the different setups, Reynolds numbers and turbulence levels in the experiments and numerical simulations, in all cases the base pressure is found to be directly proportional to the length of the mean recirculation region, which is thus a key index of the base drag value. In turn, the recirculation length seems to be connected with the location of the incipient instability of the detaching shear layers, which can be moved downstream by an increase of the thickness of the separating boundary layer and upstream by an increase of the turbulence level.

Impact of dynamic subgrid-scale modeling in variational multiscale large-eddy simulation of bluff-body flows

The effects of dynamic subgrid-scale (SGS) modeling are investigated in variational multiscale (VMS) large-eddy simulation (LES) simulations of bluff-body flows. The spatial discretization is based on a mixed finite-element/finite-volume formulation on unstructured grids. In the VMS approach used in this work, the separation between the largest and the smallest resolved scales is obtained through a variational projection operator and a finite-volume cell agglomeration. The dynamic and non-dynamic versions of Smagorinsky and wall-adapted local eddy-viscosity SGS models are used to account for the effects of the unresolved scales. In the VMS approach, these effects are only modeled in the smallest resolved scales. The VMS-LES and classical LES approaches, combined with the considered dynamic and non-dynamic SGS models, are applied to the simulation of the flow around a circular cylinder at Reynolds numbers 3,900 and 20,000 and to the flow around a square cylinder at Reynolds numbers 22,000 and 175,000.

Simulation of the flow past a circular cylinder in the supercritical regime by blending RANS and variational-multiscale LES models

A strategy which blends a variational multiscale large eddy simulation (VMS-LES) model and a RANS model in a hybrid approach is investigated. A smooth blending function, which is based on the value of a blending parameter, is used for switching from VMS-LES to RANS. Different definitions of the blending parameter are investigated. The capabilities of the novel hybrid approach are appraised in the simulation of the flow around a circular cylinder at a Reynolds number 1.4×105, based on the freestream velocity and on the cylinder diameter, in the presence of turbulent boundary-layer due to turbulent inflow conditions. A second study at Reynolds numbers from Re=6.7×105 to 1.25×106 is also presented. The effect of using the VMS-LES approach in the hybrid model is evaluated. Results are compared to those of other RANS, LES and hybrid simulations in the literature and with experimental data

Isogeometric variational multiscale large-eddy simulation of fully-developed turbulent flow over a wavy wall

We report on the isogeometric residual-based variational multiscale (VMS) large eddy simulation of a fully developed turbulent flow over a wavy wall. To assess the predictive capability of the VMS modeling framework, we compare its predictions against the results from direct numerical simulation (DNS) and large eddy simulation (LES) and, when available, against experimental measurements. We use C1 quadratic B-spline basis functions to represent the smooth geometry of the sinusoidal lower wall and the solution variables. The Reynolds numbers of the flows considered are 6760 and 30,000 based on the bulk velocity and average channel height. The ratio of amplitude to wavelength (α/λ) of the sinusoidal wavy surface is set to 0.05. The computational domain is 2λ×1.05λ×λ in the streamwise, wall-normal and spanwise directions, respectively. For the Re=6760 case, mean averaged quantities, including velocity and pressure profiles, and the separation/reattachment points in the recirculation region, are compared with DNS and experimental data. The turbulent kinetic energy and Reynolds stress are in good agreement with benchmark data. Coherent structures over the wavy wall are observed in isosurfaces of the Q-criterion and show similar features to those previously reported in the literature. Comparable accuracy to DNS solutions is obtained with at least one order of magnitude fewer degrees of freedom. For the Re=30,000 case, good agreement was obtained for mean wall shear stress and velocity profiles compared with available LES results reported in the literature.

Variational multiscale large-eddy simulations of the flow past a circular cylinder: Reynolds number effects

Variational multiscale large-eddy simulations (VMS–LES) of the flow around a circular cylinder are carried out at different Reynolds numbers in the subcritical regime, viz. Re = 3900, 10,000 and 20,000, based on the cylinder diameter. A mixed finite-element/finite-volume discretization on unstructured grids is used. The separation between the largest and the smallest resolved scales is obtained through a variational projection operator and finite-volume cell agglomeration. The WALE subgrid scale model is used to account for the effects of the unresolved scales; in the VMS approach, it is only added to the smallest resolved ones. The capability of this methodology to accurately predict the aerodynamic forces acting on the cylinder and in capturing the flow features are evaluated for the different Reynolds numbers considered. The sensitivity of the results to different simulation parameters, viz. agglomeration level and numerical viscosity, is also investigated at Re = 20,000.

Wall-stress boundary conditions for variational-multiscale LES

Two methods are introduced for the implementation of wall-stress boundary conditions in variational-multiscale large-eddy simulations. In both methods, the boundary conditions are applied weakly using an interior-penalty approach, where the discretization parameters are determined using information from a wall-stress model. The performance of the methods is compared with that of hard Dirichlet boundary conditions for turbulent channel flows. A convergence study is performed for the second of the methods, which is found to be the most viable for practical application. Sources of error affecting the convergence study are discussed and quantified. Copyright © 2010 John Wiley & Sons, Ltd.

Variational multiscale turbulence modelling in a high order spectral element method

In the variational multiscale (VMS) approach to large eddy simulation (LES), the governing equations are projected onto an a priori scale partitioning of the solution space. This gives an alternative framework for designing and analyzing turbulence models. We describe the implementation of the VMS LES methodology in a high order spectral element method with a nodal basis, and discuss the properties of the proposed scale partitioning. The spectral element code is first validated by doing a direct numerical simulation of fully developed plane channel flow. The performance of the turbulence model is then assessed by several coarse grid simulations of channel flow at different Reynolds numbers.

Development of a Variational Multiscale Large-Eddy Simulation Code ‘MISTRAL’ Using Double-Scale Finite Elements

A variational multiscale large-eddy simulation (VMS-LES) code, named MISTRAL, has been developed based upon the finite element method (FEM) for accurate and practical computation of geometrically complicated turbulent flow problems. The numerical strategy of the FEM-based VMS-LES is explained, especially focusing on the double-scale approximation for velocity and pressure in the incompressible Navier-Stokes equations, a pressure stabilization technique and a multiscale turbulence modeling. A unique technique is also employed in the time integration to realize an efficient inversion of the multiscale mass matrix and to form the multiscale pressure Poisson equation used in the approximate projection method for divergence-free constraint of velocity. As a numerical demonstration, a 2D driven cavity flow problem has been solved with the MISTRAL code in a wide range of Reynolds number (Re=1000 to 50000). The results are compared with reference data to quantitatively estimate the accuracy (magnitude of errors in terms of L2 norms) of the proposed VMS-LES method.

The variational multiscale method for laminar and turbulent flow

The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.

Variational Multiscale Large Eddy Simulation of Turbulent Flow in a Diffuser

In the present study, variational multiscale large eddy simulation (LES) is applied to turbulent flow in a diffuser, which represents a challenging test case due to the appearance of flow separation and subsequent reattachment. Two different scale-separating operators based on multigrid operators are used for separating large and small resolved scales. Dynamic as well as constant-coefficient-based subgrid-scale modeling are employed in the variational multiscale LES. The results show that the variational multiscale LES with a simple constant-coefficient Smagorinsky model using a projective scale-separating operator yields a very good prediction of most of the crucial flow features. Furthermore, it turns out to be by far the most computationally efficient approach, resulting in a computing time reduction of about 18% compared to the traditional LES with a dynamic Smagorinsky model. The variational multiscale LES with a dynamic Smagorinsky model does not prove to be a superior approach.

Scale-separating operators for variational multiscale large eddy simulation of turbulent flows

A general class of scale-separating operators based on combined multigrid operators is proposed and analyzed in this work. The operators of this class are designed for variational multiscale large eddy simulation using a finite volume or finite element method. Two representatives are compared to discrete smooth filters, which are widely used in the traditional large eddy simulation literature; the comparison shows that they are not only theoretically different, but also yield considerable differences in the respective numerical results. Dynamic as well as constant-coefficient-based subgrid-scale modeling is used within the multiscale environment. All of the scale-separating operators are implemented in a second-order accurate energy-conserving finite volume method and tested for the case of a turbulent channel flow. One operator shows particularly remarkable results in the framework of the variational multiscale large eddy simulation, that is, profiles are obtained for velocity and kinetic energy which are considerably closer to the respective profiles from a direct numerical simulation than are the profiles resulting from the application of the other operators considered in the present study. Furthermore, this particular operator proves to be very efficient with regard to the important aspect of computational cost, that is, a reduction in computing time ranging from about 25% up to about 150% compared to the other operators. The introduction of a substantial amount of subgrid viscosity to the small scales, particularly in the buffer layer of the channel, appears to be crucial for the good results achieved with this method.

Energy transfers and spectral eddy viscosity in large-eddy simulations of homogeneous isotropic turbulence: Comparison of dynamic Smagorinsky and multiscale models over a range of discretizations

Energy transfers within large-eddy simulation (LES) and direct numerical simulation (DNS) grids are studied. The spectral eddy viscosity for conventional dynamic Smagorinsky and variational multiscale LES methods are compared with DNS results. Both models underestimate the DNS results for a very coarse LES, but the dynamic Smagorinsky model is significantly better. For moderately to well-refined LES, the dynamic Smagorinsky model overestimates the spectral eddy viscosity at low wave numbers. The multiscale model is in good agreement with DNS for these cases. The convergence of the multiscale model to the DNS with grid refinement is more rapid than for the dynamic Smagorinsky model.

Sensitivity of the scale partition for variational multiscale large-eddy simulation of channel flow

The variational multiscale method has been shown to perform well for large-eddy simulation (LES) of turbulent flows. The method relies upon a partition of the resolved velocity field into large- and small-scale components. The subgrid model then acts only on the small scales of motion, unlike conventional LES models which act on all scales of motion. For homogeneous isotropic turbulence and turbulent channel flows, the multiscale model can outperform conventional LES formulations. An issue in the multiscale method for LES is choice of scale partition and sensitivity of the computed results to it. This is the topic of this investigation. The multiscale formulation for channel flows is briefly reviewed. Then, through the definition of an error measure relative to direct numerical simulation (DNS) results, the sensitivity of the method to the partition between large- and small-scale motions is examined. The error in channel flow simulations, relative to DNS results, is computed for various partitions between large- and small-scale spaces, and conclusions drawn from the results.