Direct numerical simulation of compressible turbulent channel flows using the discontinuous Galerkin method

Direct numerical simulation of turbulent channel flows between isothermal walls have been carried out using discontinuous Galerkin method. Three Mach numbers are considered (0.2, 0.7, and 1.5) at a fixed Reynolds number ≈2800, based on the bulk velocity, bulk density, half channel width, and dynamic viscosity at the wall. Power law and log-law with the scaling of the mean streamwise velocity are considered to study their performance on compressible flows and their dependence on Mach numbers. It indicates that power law seems slightly better and less dependent on Mach number than the log-law in the overlap region. Mach number effects on the second-order (velocity, pressure, density, temperature, shear stress, and vorticity fluctuations) and higher-order (skewness and flatness of velocity, pressure, density, and temperature fluctuations) statistics are explored and discussed. Both inner (that is wall variables) and outer (that is global) scalings (with Mach number) are considered. It is found that for some second-order statistics (i.e. velocity, density, and temperature), the outer scaling collapses better than the inner scaling. It is also found that near-wall large-scale motions are affected by Mach number. The near-wall spanwise streak spacing increases with increasing Mach number. Iso-surfaces of the second invariant of the velocity gradient tensor are more sparsely distributed and elongated as Mach number increases, which is similar to the distribution of near-wall low speed streaks.     

A numerical investigation of turbulent flows in a spanwise rotating channel

A numerical investigation of fully developed turbulent flow in a spanwise rotating channel is performed to study turbulence characteristics subject to system rotation. The work provides insight into several salient features of the spanwise rotating turbulent channel flows, including the near-wall vortical structures, turbulence energy cascade and redistribution, and vortex stretching. The influence of system rotation on the near-wall vortical structures is investigated based on the vorticity fluctuations and their probability density functions (PDF). The properties of the Lamb vector fluctuation and the corresponding PDF are examined to reveal the effect of rotation on the turbulence energy cascade and production in the rotating channel. The budgets of Reynolds stresses and fluctuating enstrophy are analyzed to elucidate the role of the Coriolis force on turbulence energy redistribution between the streamwise and wall-normal directions and the mechanisms of vortex stretching for the generation of the vorticity fluctuations near the pressure and suction walls.     

An investigation of pulsating turbulent open channel flow by large eddy simulation

Pulsating turbulent open channel flow is investigated by use of large eddy simulation (LES) technique coupled with a dynamic subgrid-scale (SGS) model for turbulent SGS stress. The three-dimensional filtered Navier-Stokes equation is numerically solved by a fractional-step method. The objective of this study is to deal with the behaviors of pulsating turbulent open channel flow, in particular turbulence characteristics in the free surface-influenced layer, and to examine the reliability of the LES approach for predicting the pulsating turbulent flow with a free surface. In this study, the frequency of driving pressure gradient ranges low, medium and high value. The mean and phase-averaged statistical turbulence quantities, the resolved turbulent kinetic energy and Reynolds stresses budgets, and the flow structures are obtained and analyzed. With the increase of the driving frequency, the depth of the surface-influenced layer increases and the turbulent Stokes length near the bottom wall decreases. Different turbulence characteristics between the accelerating and decelerating phases are interpreted comprehensively. Turbulence intensities reveal that turbulent flow has a strong anisotropy in the free surface-influenced layer, in particular in the decelerating phases during the pulsating cycle. The budget terms of the resolved turbulent kinetic energy, the vertical and spanwise Reynolds stresses in the free surface region are analyzed. The flow structures clearly exhibit that bursting processes near the bottom wall are ejected toward the surface and the most surface renewal events are closely correlated with the bursting processes. These processes are strengthened during the decelerating period since strong turbulence intensities are generated.     

The Reynolds stress vector and tensor

The problem of the asymmetry of the stress tensor is considered. The mechanism of the origination of turbulence in a flow is shown. On the basis of a vector form of the Newton’s law for the viscous liquid, a tensor of the generalized viscosity coefficient whose components take into account both the molecular and turbulent viscosity is introduced. It is shown that the Reynolds turbulent stresses can be represented as a Reynolds stress vector. An expression is found for the Reynolds stresses in the boundary layer approximation.     

A parallel, finite-volume algorithm for large-eddy simulation of turbulent flows

A parallel, finite-volume algorithm has been developed for large-eddy simulation (LES) of compressible turbulent flows. This algorithm includes piecewise linear least-square reconstruction, trilinear finite-element interpolation, Roe flux-difference splitting (FDS), and second-order MacCormack time marching. A systematic and consistent means of evaluating the surface and volume integrals of the control volume is described. Parallel implementation is done using the message-passing programming model. To validate the numerical method for turbulence simulation, LES of fully developed turbulent flow in a square duct is performed for a Reynolds number of 320 based on the average friction velocity and the hydraulic diameter of the duct. Direct numerical simulation (DNS) results are available for this test case, and the accuracy of this algorithm for turbulence simulations can be ascertained by comparing the LES solutions with the DNS results. For the first time, a finite volume method with Roe FDS was used for LES of turbulent flow in a square duct, and the effects of grid resolution, upwind numerical dissipation, and subgrid-scale dissipation on the accuracy of the LES are examined. Comparison with DNS results shows that the standard Roe FDS adversely affects the accuracy of the turbulence simulation. For accurate turbulence simulations, only 3–5% of the standard Roe FDS dissipation is needed.     

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.     

On the realizability of reynolds stress turbulence closures

The realizability of Reynolds stress models in homogeneous turbulence is critically assessed from a theoretical standpoint. It is proven that a well known second-order closure model formulated using the strong realizability constraints of Schumann (1977) and Lumley (1978) is, in fact, not a realizable model. The problem arises from the failure to properly satisfy the necessary positive second time derivative constraint when a principal Reynolds stress vanishes-a flaw that becomes apparent when the nonanalytic terms in the model are made single-valued as required on physical grounds. More importantly, arguments are advanced which suggest that it is impossible to identically satisfy the strong from of realizability in any version of the present generation of second-order closures. On the other hand, models properly formulated to satisfy the weak form of realizability—wherein states of one or two component turbulence are made inaccessible in finite time via the imposition of a positive first derivative condition—are found to be realizable. However, unlike the simpler and more commonly used second-order closures, these models can be ill-behaved near the extreme limits of realizable turbulence due to the way that higher-degree nonlinearities are often unnecessarily introduced to satisfy realizability. Illustrative computations of homogeneous shear flow are presented to demonstrate these points which can have important implications for turbulence modeling.     

Explicit algebraic Reynolds stress model of incompressible turbulent flow in rotating square duct

An explicit algebraic Reynolds stress model (EARSM) is proposed for the simulation of the incompressible three-dimensional Reynolds averaged Navier-Stokes (RANS) equations. The spatial discretization of the RANS equations is performed by a finite volume method with nonstaggered variable arrangement.The EARSM model which accounts for rotational effects is used to compute the turbulent flows in rotating straight square duct. The Reynolds number of 48,000 is based on the bulk velocity and the hydraulic diameter of the duct and is kept constant in the range of the rotational numbers. The second order closure (EARSM) yields an asymmetric mean velocity profile as well as turbulence properties. Effects of rotation near the cyclonic (suction side) and anticyclonic (pressure side) walls are well observed. Direct numerical simulation and large eddy simulation data are available for this case. The comparison of EARSM results with these accurate simulations shows a very good agreement.     

A Combined Large-Eddy Simulation and Time-Dependent RANS Capability for High-Speed Compressible Flows

An entirely new approach to the large-eddy simulation (LES) of high-speed compressible turbulent flows is presented. Subgrid scale stress models are proposed that are dimensionless functions of the computational mesh size times a Reynolds stress model. This allows a DNS to go continuously to an LES and then a Reynolds-averaged Navier–Stokes (RANS) computation as the mesh becomes successively more coarse or the Reynolds number becomes much larger. Here, the level of discretization is parameterized by the nondimensional ratio of the computational mesh size to the Kolmogorov length scale. The Reynolds stress model is based on a state-of-the-art two-equation model whose enhanced performance is documented in detail in a variety of benchmark flows. It contains many of the most recent advances in compressible turbulence modeling. Applications to the high-speed aerodynamic flows of technological importance are briefly discussed.     

Towards development of unsteady near-wall interface boundary conditions for turbulence modeling

In near-wall turbulence modeling it is required to resolve a thin layer nearby the solid boundary, which is characterized by high gradients of the solution. An accurate enough resolution of such a layer can take most computational time. The situation even becomes worse for unsteady problems. To avoid time-consuming computations, a new approach is developed, which is based on a non-overlapping domain decomposition. The boundary condition of Robin type at the interface boundary is achieved via transfer of the boundary condition from the wall. For the first time interface boundary conditions of Robin type are derived for a model nonstationary equation which simulates the key terms of the unsteady boundary layer equations. In the case of stationary solutions the approach is automatically reduced to the technique earlier developed for the steady problems. The considered test cases demonstrate that unsteady effects can be significant for near-wall domain decomposition. In particular, they can be important in the case of the wall-function-based approach.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

Studies of shock/turbulent shear layer interaction using Large-Eddy Simulation

Shock/shear/turbulence interactions are simulated using Large-Eddy Simulation (LES) with a new localized subgrid closure approach. Both normal and oblique shocks interactions with turbulence are considered. The LES methodology adopted here combines a hybrid numerical scheme that switches automatically and locally between a shock-capturing scheme and a low-dissipation high-order central scheme.The fundamental role of the diffusion of turbulent kinetic energy by pressure fluctuations in the problem of normal shock/isotropic turbulence interaction is stressed in the DNS study, and accounted for in the closure model. The study of the interaction between two oblique shocks and a turbulent shear layer shows that the turbulence evolution is mostly affected by two competing phenomena. An amplification of the turbulent levels occurs downstream of the interaction, and the mixing layer growth rate is significantly increased. However, the integrated production of turbulent energy across the mixing layer is reduced, and the increase in mixing is found to be localized in space, the turbulent statistics quickly relaxing to their undisturbed levels. Furthermore, the increase in vorticity from the compression of the mixing layer remains small, unaffected by the presence of turbulent and coherent structures.     

Analytical solution of mixed electromagnetic/pressure driven gaseous flows in microchannels

The influences of wall-slip/jump conditions on the fluid flow and heat transfer for hydrodynamically and thermally fully developed electrically conducting gaseous flow subject to an electromagnetic field inside a parallel plate microchannel with constant heat flux at walls are studied under the assumptions of a low-magnetic Reynolds number. The governing equations are non-dimensionalized and then analytical solutions are derived for the friction and the heat transfer coefficients. The fluid flow and the heat transfer characteristics obtained in the analytical solutions are discussed in detail for different parameters such as the Knudsen, Hartmann, and Brinkman numbers. The velocity profiles verify that even with a constant Knudsen number, applying a stronger electromagnetic field gives rise to an increase in the slip velocity. The results also reveal that on increasing the Hartmann number, the heat transfer rate as well as the friction factor is enhanced, whereas it tends to suppress the movement of the fluid. Further, it is found that the Nusselt and the Poiseuille numbers are less sensitive to the electromagnetic field effects with increase in rarefaction.     

Semi-analytical solutions of pulsating flow in a helically rough-walled microtube

The influence of a helically rough wall surface on fully developed pulsating laminar flows in microtube is investigated through a semi-analytical method. A two-dimensional simple harmonic function is chosen to model helically wavy structure. The velocity and pressure are decomposed into space-averaged and disturbance terms and the corresponding governing equations are established. Pulsating flow is split into a steady term and an oscillatory term based on a given oscillatory pressure drop, and the governing equations for the oscillatory terms are presented together with their steady counterparts. Analytical solutions are obtained for the space-averaged equations and a spectral collocation method is used to solve the disturbance equations numerically. An iterative approach is adopted for the coupled equations with respect to space-averaged velocities and disturbance velocities. The computational results show that a Reynolds stress layer (RSL) and a secondary swirling velocity are present in the rough-walled microtube. The relative amplitude of the waviness of the wall, its wavenumber, and the Reynolds number are found to be important parameters influencing the Reynolds stress and RSL, as well as the space-averaged velocities and pressure drop. The direction of rotation of the swirling velocity is found to depend on the sign of the azimuthal wavenumber. In addition to the parameters of the rough wall and the Reynolds number, both the oscillatory pressure drop and frequency are important factors influencing the pulsating flow and the phase shift of the space-averaged velocity.     

A computational study of wall friction and turbulence dynamics in accelerating pipe flows

A CFD model of turbulent flow in a smooth pipe accelerating uniformly from steady state is used to study the influence of turbulence and inertia on wall shear stresses. A low-Reynolds-number k-ε turbulence model is used in conjunction with a finite volume/finite difference discretization scheme. It is shown that the wall shear stress initially overshoots the corresponding quasi-steady value and this is attributed to inertial causes. Thereafter, the wall shear stress is shown to undershoot the quasi-steady value because inertial effects are more than counterbalanced by the cumulative influence of delays in the response of turbulence to flow changes. The dependence of the flow behaviour on the geometry, the fluid properties, the Reynolds number and the acceleration is studied and is shown to correlate well with a non-dimensional parameter based on the turbulence production timescale. The durations of the initial overshoots and the amplitudes of the overshoots and undershoots are smaller at high Reynolds numbers than at low ones.     

Turbulent boundary layer structure of flow over a smooth-curved ramp

The effects of a convex-curved wall followed by a recovery over a flat surface on a turbulent boundary layer structure are addressed via large-eddy simulation (LES). The curved wall constitutes a smooth ramp formed by a portion of circular arc. The statistically two-dimensional upstream boundary layer flow is realistically fed by an injected inflow boundary condition. The inflow is extracted from a simultaneously simulated flat-plate boundary layer which is computed based on a compressible rescaling method. After flowing over the curved surface the flow is allowed to recover its realistic condition by passing over a downstream flat surface. The Reynolds number introduced at the inlet section of the computational domain which starts 4 times the ramp length (L_r) upstream of the curved surface is Re_δo=U_∞δ_o/ν=9907. The Reynolds number is based on the inflow boundary layer thickness δ_o, the free-stream velocity U_∞ and the kinematic viscosity ν.Mean flow predictions obtained using the present LES with the rescaling–recycling inflow condition agree well with the available experimental data from literature. The Reynolds stress components match the experimental one. However, small deviation occurs due to the smaller-domain height used in the present simulation. The experiments showed that there is a generated pressure gradient on the upper wall and this in return affects the turbulence energy on the other wall. The numerical data as well as the experiments show an enhancement of the turbulent stresses in the adverse pressure gradient region. The increased level of turbulent stresses is accompanied with large peaks aligned with the inflection point of the velocity profiles. The high stress levels are nearly unchanged by reattachment process, decaying only after the mean velocity recovered and the high production of turbulence near the outer layer drops. The recovery of the outer layer is due to the turbulent eddies generated by the separation region. Numerical visualizations show strong elongation and lifting of eddies in the region of the adverse pressure gradient generated by the curved wall. Computations of two-point correlations are also performed to represent the formation and deformation of the turbulent eddies before, over and after the curved wall. Different effects on the eddy size and its structure angle are presented.     

CFD modeling of scale effects on turbulence flow and scour around bridge piers

Sediment scour near bridge piers is a problem of nationwide concern because it has resulted in more bridge failures than all other causes in recent years. The existing bridge scour equation from HEC-18 was developed from laboratory experiments in relatively small scale. Field studies by Mueller [Mueller D, Wagner Chad R. Analysis of pier scour predictions and real-time field measurements. In: Proceedings of ICSF-1 first international conference on scour of foundations, Texas A&M University, College Station, Texas, USA; 2002] indicate that it is difficult to verify the scour equation with field data obtained from large bridge piers. In this study, computational model simulations using a 3D CFD model were conducted to examine scale effects on turbulent flow and sediment scour. For the large-scale model, the physical scale and boundary velocity were set up from the small scale model based on the Froude similarity law. Results of flow and sediment scour were obtained from two different approaches: (a) Froude similarity which is commonly used in physical modeling and (b) full scale 3D CFD modeling. Unlike physical modeling in which the effect of turbulent Reynolds number is ignored, the CFD model employs a 2nd order turbulent model to calculate turbulent velocity and sediment scour. Effects of scale on turbulence flow and sediment scour were investigated by comparing different results obtained from a full scale numerical model to those derived from the Froude similarity method.     

Instability of channel flow with fluid injection and parietal vortex shedding

This paper is devoted to the computation of the stability properties and the aeroacoustic interaction in a planar channel flow driven by air injection through porous wall. The purpose is to establish a simulation method which estimates the vortex-shedding and acoustic-wave interaction. To perform these unsteady computations, turbulent motion is taken into account by a first-order model based on a nonlinear relationship between time-dependent Reynolds stresses and velocity gradients. Model coefficients are explicit functions of both strain and rotation. This model is applied to the computation of parietal vortex shedding in a simplified configuration. It is a channel flow with fluid injection. Two cases are computed here. The first one is stable and the second one is unstable. To improve the evaluation of the turbulent effects, computations of the second case are presented with and without a turbulence model. In first configuration, a good agreement with experimental data is found. In the second case both with and without turbulence model the natural unsteadiness of the flow is captured. The parietal vortex shedding is described and the turbulence effects are characterized.     

Unsteady turbulence in plane channel flow

Direct numerical simulations were conducted for oscillating flow with zero time mean (reciprocating flow) in a plane channel subject to a harmonic forcing term of varying amplitude and frequency. The results confirmed the existence of four flow regimes (laminar, “disturbed laminar”, intermittently turbulent, and fully turbulent) depending on the above parameters. The flow behaviour was found to depend on the complex interplay of mean and turbulence quantities, as described by the closed loop formed by the streamwise Reynolds-averaged momentum equation in conjunction with the exact transport equations for the turbulent (Reynolds) stresses. A crucial role in this loop appeared to be played by the different time response of the mean flow to the applied forcing at different cross-stream locations, due to the laminar and turbulent diffusion of momentum from the walls and causing characteristic distortions of the cross-stream mean velocity profiles at different phases (i.e. acceleration vs. deceleration). The intrinsic inertia of turbulence quantities themselves played only a minor role, as confirmed by the fact that, in a broad range of conditions, turbulent stresses were roughly in phase with the respective production and dissipation terms. The structure of turbulence was found to depend largely on the instantaneous mean velocity profile, as confirmed by “frozen velocity” simulations.     

Enhanced injection method for synthetically generated turbulence within the flow domain of eddy-resolving simulations

The quality of eddy-resolving turbulence simulations strongly depends on appropriate inflow conditions. In most cases they have to be time-dependent and satisfy certain conditions for the first (mean velocities) and second-order moments (Reynolds stresses) as well as concerning suitable length scales. To mimic a physically realistic incoming flow, synthetically generated turbulent velocity fluctuations superimposed on the mean velocity field are a valuable solution. However, the resolution of the grid near the inlet has to be sufficiently fine to avoid excessive damping of the turbulence intensity. In order to circumvent this problem, the injection of synthetically generated inflow data not at the inlet itself but inside the flow domain near the area of interest, where the grid is typically much finer, is an elegant loophole. In the present study two different injection techniques based on a source-term formulation are analyzed and evaluated. In addition to these techniques the injected data are weighted by a Gaussian distribution defining the influence area. In the recent work the definition of the influence area is enhanced compared to the initial version of Schmidt and Breuer (2017) extending the application range. The case of a rather small influence area in comparison with the grid cell size is now tackled which is often relevant for industrial applications.The flow past a wall-mounted hemisphere is chosen as test case. The bluff body is exposed to a thick turbulent boundary layer at Re $=$ 50,000. The generation of the turbulent velocity fluctuations in the present investigation relies on the digital filter concept, but the injection techniques evaluated are not restricted to this inflow generator. The synthetic turbulent velocity fluctuations are injected about one diameter upstream of the hemisphere. Wall-resolved large-eddy simulations are carried out for two grid resolutions and the corresponding results are analyzed and compared with the reference measurements by Wood et al. (2016). Finally, one injection technique is found to be clearly superior to the other, since it guarantees the correct level of the velocity fluctuations and the reproduction of the autocorrelations.