Elliptic Relaxation Equation Research Articles

Simulation of Pressure–Velocity Correlations by Green’s Function Based on Reynolds Stress Model

Cost-effective wind energy harvesting by wind turbines in urban areas needs to strengthen the required flow field properties, such as mean velocity, turbulence, and its distribution. This paper conducts a series of CFD simulations to investigate the characteristics and related mechanisms of flow within the cavity, considering the force–turbulence interactions at the RANS scales. The pressure–velocity correlation term is formulated and solved by the elliptic relaxation equation to compensate for the Reynolds stress overestimation. Numerical simulations of flow over an open cavity with the proposed model are compared with corresponding PIV data. The results show that the mean velocity and the fluctuation velocity along the streamwise direction exist a slightly favorable pressure gradient. While the fluctuation velocity and fluctuation pressure show different correlation characteristics along the streamwise direction. Moreover, the pressure–velocity fluctuation correlation becomes obvious near the upper corner of the cavity due to the favorable pressure gradient. Hence, the leading and trailing locations of the cavity are both obvious favorable regions and further emphasis should be put on both high-accurate simulation methods and practical applications.

Elliptic Blending with One-Equation Model

A wall-distance-free modification to Wray-Agarwal (WA) one-equation turbulence model is convoked using an elliptic relaxation approach to accurately accounting for non-local characteristics of near-wall turbulence. Model coefficients/functions are parameterized with the elliptic relaxation function to preserve the combined effects of near-wall turbulence and nonequilibrium. The characteristic length scale associated with the elliptic relaxation equation is formulated in terms of viscous and turbulent length scales in conjunction with the invariant of strain-rate tensor. Consequently, non-local effects are explicitly influenced by the mean flow and turbulent variables. A near-wall damping function is introduced to relax the viscous length-scale coefficient adhering to the elliptic relaxation model. Comparisons indicate that the new model improves the accuracy of flow simulations compared to the widely used Spalart-Allmar as model and remains competitive with the SST model.A good correlation is obtained between the current model and DNS/experimental data.

Numerical study of an impinging jet in cross-flow within and without influence of vortex generator structures on heat transfer

An impinging jet heat transfer in cross-flow within and without influence of a vortex generator pair (VGP) is studied using the unsteady Reynolds averaged Navier-Stokes (URANS) and the large-eddy simulation (LES). The jet Reynolds number is 15,000 and the cross-flow Reynolds number is 30,000. The elliptic-blending Reynolds stress model (EBRSM) is implemented and adapted to capture the effect of the jet close to the wall. A v2 − f model is also implemented to study the ability in predicting such a benchmark. Both models benefit from the elliptic relaxation equation in the entire computational domain. The URANS results are compared with the accurate results of the LES method and also the experimental data. The URANS method successfully presents the flow features of the impinging jet while underpredicts the enhancing heat transfer over the channel bottom wall. The URANS method fails to correctly predict the flow structures forming around the impinging region, because the method is more diffusive than the LES method. When manipulating VGP, a rectangular winglet vortex generator pair is placed in the cross-flow channel and upstream of the jet nozzle to enhance the impinging heat transfer. The VGP increases the Nusselt number at the impingement region. The structures generated by the VGP alter the effects of the cross-flow on the impinging heat transfer. There are Kelvin-Helmholtz instabilities at the shear layer of the jet and the cross-flow in the base flow (the flow without VGP). These instabilities are altered in the flow with VGP. A swirl component is added in the jet to study the effects on the heat transfer. The result shows that for a high or moderate level of swirl, the jet is diffused before the impinging.

Elliptic relaxation model for stably stratified turbulence

The article presents a new elliptic relaxation procedure for Second Moment Closure (SMC) turbulence modeling of wall-bounded, stably stratified flows. Specifically, it directly introduces buoyancy effects in the elliptic relaxation equation. The new formulation results in an additional length scale that produces a coupled set of Helmholtz equations. The study then extends the new technique to Reynolds scalar fluxes. Furthermore, it examines the budgets of the different terms of their transport equations near the wall and systematically derives the relevant boundary conditions. Comparison of model predictions with Direct Numerical Simulation (DNS) data for stably stratified plane channel flows available in the literature rounds off this paper.

On eddy viscosity transport models with elliptic relaxation

ABSTRACTAn eddy viscosity transport (EVT) model is formulated based on the transformation of the standard k–ϵ model without the assumption of equal diffusion coefficients. These terms that are usually neglected, due to their complexities and numerical difficulties, are regrouped so that an elliptic production term naturally emerges. This term forms the basis of an elliptic relaxation model that avoids the need for wall-damping functions. All of the EVT model's constants follow from the transformation, maintaining the closest relationship to its parent k–ϵ model. To assess the ability of the elliptic relaxation equation to account for the non-local wall effects, another version of the EVT model that neglects the diffusion terms and retains only the Laplace operator term, i.e., , is proposed and compared to Menter one-equation model, which utilises wall-damping functions. The performance of this version of the EVT model is also compared to the fully transformed EVT model in order to assess the importance of the neglected diffusion terms. All three models are tested in detail over a wide class of internal and external turbulent flows.

Detached-eddy simulation based on the [formula omitted] model

Detached-eddy simulation (DES) based on the v2-f Reynolds-averaged Navier–Stokes (RANS) model is developed and tested. The v2-f model incorporates anisotropy of near-wall turbulence, which is absent in other RANS models commonly used for DES. The v2-f RANS model is modified in order that in the DES formulation it reduces to a transport equation for the subgrid-scale kinetic energy in isotropic turbulence. First, three coefficients in the elliptic relaxation equation are modified and the modification is tested for RANS of channel flow. Next, the proposed v2-f DES model is formulated. The constant, CDES, required in the DES formulation is calibrated by simulating both decaying and statistically-steady isotropic turbulence. The formulation is tested for flow around a circular cylinder at a Reynolds number of 3900, in which case turbulence develops after separation. Simulations indicate that this model represents the turbulent wake nearly as accurately as the dynamic Smagorinsky model. For comparison, Spalart–Allmaras (SA) based DES is also included in the cylinder flow simulations. Finally, the proposed model is tested for a separated turbulent boundary layer on an airfoil. The delayed-DES (DDES) approach is found to be necessary. The v2-f DDES calculation gives good predictions for the aerodynamic forces and the pressure distribution measured in experiments. The proposed model is also compared with SA-DDES for the airfoil.

Amplification effect of redistribution from elliptic relaxation equation with Reynolds stress model in rotating flow

A new elliptic relaxation operator that is able to correct the erroneous increase of the redistribution from the original one suggested by Durbin [J. Fluid Mech. 249], is developed by using an inhomogeneous correction to its source term. In this work the amplification of the redistribution arising from the original model is investigated in detail by using DNS analysis. It is shown that the amplification effect extends the near-wall layer into the wake layer as well as the logarithmic layer. Also, it can be seen that the remedy of all modified neutral operators proposed so far is limited only in the logarithmic layer. However, the behavior of the present operator can be significantly improved in the overall region without amplification of the redistribution. The computational results for the new operator agree quite well with the DNS/LES data without using any ad hoc additional term in the governing equations in order to obtain a rotating effect.

Numerical Investigation of Steady Density Currents Flowing Down an Incline Using v2¯−f Turbulence Model

The governing equations of two-dimensional steady density currents are solved numerically using a finite volume method. The v2¯−f turbulence model, based on standard k−ε model, is used for the turbulence closure. In this method, all Reynolds stress equations are replaced with both a transport equation for v2¯ and an elliptic relaxation equation for f, a parameter closely related to the pressure strain redistribution term. The Simple-C procedure is used for pressure-velocity coupling. In addition, Boussinesq’s approximation is used to obtain the momentum equation. The computed height of the progressive density current is compared to the measured data in the literature, resulting in good agreement. The present results show that the flow rate is the most dominant parameter among those affecting the density currents hydrodynamics. The results also show that the v2¯−f turbulence model is able to predict and simulate the characteristics of the low Reynolds turbulent density currents successfully, although it is based on a high Reynolds number turbulence model, i.e., the standard k−ε model. The use of boundary layer convention, saying that the density current’s height is a height at which the concentration is ∼1% of the inlet concentration, seems to yield reasonable results.

An alternative two-equation turbulent heat diffusivity closure

The objective of this paper is to present an alternative closure for turbulent heat transport, that has been established on the ground of v 2– f model with so-called elliptic relaxation equation and the two-equation model for turbulent thermal field. New formulae on the eddy heat diffusivity is proposed. Additionally the production term of destruction rate of temperature variance transport equation is reformulated. No damping functions are involved in this model. Results of numerical computation have been compared with the experimental data for fully developed thermal field in the pipe and DNS heat transfer prediction for two-dimensional channel flow. The model predicts reasonably well the near-wall distribution of basic turbulence statistics.

Inhomogeneity and anisotropy effects on the redistribution term in Reynolds-averaged Navier–Stokes modelling

A channel flow DNS database at Reτ = 590 is used to assess the validity of modelling the redistribution term in the Reynolds stress transport equations by elliptic relaxation. The model assumptions are found to be globally consistent with the data. However, the correlation function between the fluctuating velocity and the Laplacian of the pressure gradient, which enters the integral equation of the redistribution term, is shown to be anisotropic. It is elongated in the streamwise direction and strongly asymmetric in the direction normal to the wall, in contrast to the isotropic, exponential model representation used in the original elliptic relaxation model. This discrepancy is the main cause of the slight amplification of the energy redistribution in the log layer as predicted by the elliptic relaxation equation. New formulations of the model are proposed in order to correct this spurious behaviour, by accounting for the rapid variations of the length scale and the asymmetrical shape of the correlation function. These formulations do not rely on the use of so-called ‘wall echo’ correction terms to damp the redistribution. The belief that the damping is due to the wall echo effect is called into question through the present DNS analysis.

Computations of transonic flow with the v2–f turbulence model

Abstract The v2–f (or k–e– v ′2 ) model of Durbin, which has shown very good performance in several test cases with separated flows at low speeds, has been applied to transonic flows over the Delery bump and the RAE 2822 airfoil. In both cases, shock-induced separation occurs. The elliptic relaxation equation of the model is used in the entire computational domain, i.e., it is computed in both the subsonic and supersonic regimes. The predicted size of the separation zone, surface pressures and mean velocities by two model variants agree reasonably well with the experimental data. Consistent with our earlier investigation, in which the v2–f model was applied to a subsonic high-lift airfoil at a high incidence angle, the level of turbulent shear–stress within the separation zone is underestimated. A `code-friendly' variant of the model is proposed to enhance numerical stability. It has proven to be advantageous if an implicit segregated equation-by-equation approach is used. This modification alleviates the `stiffness problem' associated with the original model caused by the boundary conditions at walls. This is particularly true when the Reynolds number is high and the near-wall grid spacing is extremely small.

A new form of the elliptic relaxation equation to account for wall effects in RANS modeling

Different methods for improving the behavior in the logarithmic layer of the elliptic relaxation equation, which enable the extension of Reynolds stress models or eddy viscosity models down to the wall, are tested in a channel flow at Reτ=590 and compared with direct numerical simulation (DNS) data. First, a priori tests are performed in order to confirm the improvement predicted by the theory, either with the Rotta+IP (isotropization of production) model or the Speziale–Sarkar–Gatski (SSG) model as the source term of the elliptic relaxation equation. The best form of the model is then used for full simulations, in Durbin second moment closure or in the frame of the v2¯−f model. It is shown that the results can be significantly improved, in particular by using a formulation based on the refinement of the modeling of the two-point correlations involved in the redistribution term.

A nonlinear low-Reynolds number heat transfer model for turbulent separated and reattaching flows

A nonlinear low-Reynolds number heat transfer model is developed to predict turbulent flow and heat transfer in separated and reattaching flows. The k– ε– f μ model of Park and Sung (T.S. Park, H.J. Sung, A new low-Reynolds-number model for predictions involving multiple surface, Fluid Dynamics Research 20 (1997) 97–113) is extended to a nonlinear formulation, based on the nonlinear model of Gatski and Speziale (G.B. Gatski, C.G. Speziale, On explicit algebraic stress models for complex turbulent flows, J. Fluid Mech. 254 (1993) 59–78). The limiting near-wall behavior is resolved by solving the f μ elliptic relaxation equation. An improved explicit algebraic heat transfer model is proposed, which is achieved by applying a matrix inversion. The scalar heat fluxes are not aligned with the mean temperature gradients in separated and reattaching flows; a full diffusivity tensor model is required. The near-wall asymptotic behavior is incorporated into the f λ function in conjunction with the f μ elliptic relaxation equation. Predictions of the present model are cross-checked with existing measurements and DNS data. The model performance is shown to be satisfactory.

A Non-linear Low-Reynolds-Number Heat Transfer Model for Turbulent Separated and Reattaching Flows

A nonlinear low-Reynolds number heat transfer model is developed to predict turbulent flow and heat transfer in separated and reattaching flows. The k‐e‐fm model of Park and Sung (T.S. Park, H.J. Sung, A new low-Reynoldsnumber model for predictions involving multiple surface, Fluid Dynamics Research 20 (1997) 97‐113) is extended to a nonlinear formulation, based on the nonlinear model of Gatski and Speziale (G.B. Gatski, C.G. Speziale, On explicit algebraic stress models for complex turbulent flows, J. Fluid Mech. 254 (1993) 59‐78). The limiting nearwall behavior is resolved by solving the fm elliptic relaxation equation. An improved explicit algebraic heat transfer model is proposed, which is achieved by applying a matrix inversion. The scalar heat fluxes are not aligned with the mean temperature gradients in separated and reattaching flows; a full diAusivity tensor model is required. The nearwall asymptotic behavior is incorporated into the fl function in conjunction with the fm elliptic relaxation equation. Predictions of the present model are cross-checked with existing measurements and DNS data. The model performance is shown to be satisfactory. # 2000 Elsevier Science Ltd. All rights reserved.

Eddy viscosity transport model for turbulent flow

An eddy viscosity transport model that is integrable to no-slip walls is described. The transport equation accommodates significant departure of the turbulence from equilibrium. The model uses an elliptic relaxation equation to avoid a need for damping functions; nonlocal wall effects are produced by solving the elliptic equation. Length scales are determined locally, without reference to distance to a surface. The model bifurcates from a laminar to a turbulent solution at a reasonable critical Reynolds number in plane channel flow. An assessment of the model is provided via boundary layer computations: these show that it behaves well in zero and adverse pressure gradients, although premature separation occurs in a strong adverse pressure gradient. Good results are obtained for the trailing edge separation bubble of an aerofoil at large angle of attack.

A Reynolds stress model for near-wall turbulence

A tensorially consistent near-wall second-order closure model is formulated. Redistributive terms in the Reynolds stress equations are modelled by an elliptic relaxation equation in order to represent strongly non-homogeneous effects produced by the presence of walls; this replaces the quasi-homogeneous algebraic models that are usually employed, and avoids the need for ad hoc damping functions. A quasi-homogeneous model appears as the source term in the elliptic relaxation equation-here we use the simple Rotta return to isotropy and isotropization of production formulae. The formulation of the model equations enables appropriate boundary conditions to be satisfied. The model is solved for channel flow and boundary layers with zero and adverse pressure gradients. Good predictions of Reynolds stress components, mean flow, skin friction and displacement thickness are obtained in various comparisons to experimental and direct numerical simulation data. The model is also applied to a boundary layer flowing along a wall with a 90°, constant-radius, convex bend. Because the model is of a general, tensorially invariant form, special modifications for curvature effects are not needed; the equations are simply transformed to curvilinear coordinates. The model predicts many important features of this flow. These include: the abrupt drop of skin friction and Stanton number at the start of the curve, and their more gradual recovery after the bend; the suppression of turbulent intensity in the outer part of the boundary layer; a region of negative (counter-gradient) Reynolds shear stress; and recovery from curvature in the form of a Reynolds stress ‘bore’ propagating out from the surface. A shortcoming of the present model is that it overpredicts the rate of this recovery. A heat flux model is developed. It is shown that curvature effects on heat transfer can also be accounted for automatically by a tensorially invariant formulation.