Multiscale Turbulence Modeling Research Articles

Validation study of a Spatially-Averaged Two-Fluid Model for heat transport in gas-particle flows

We validate a multi-scale turbulence model for the thermal energy estimation in coarse-grid simulations of moderately dense gas-particle flows. The model is based on spatially-averaging the Two-Fluid Model balance equations including the thermal energy balance equation of both phases. We found, that the drift temperature is a valid measure for the heat transfer reduction due to heterogeneous particle clusters. Furthermore, we propose the dynamic estimation of the drift temperature through the application of test-filters and the solution of transport equations for the variances of the phase temperatures and the solid volume fraction. Closure models for the Reynolds stress contributions are based on single-phase Large-Eddy Simulation models, such as gradient assumptions. In this study, we consider different test-cases including Geldart type A and type B particles in moderately dense regimes with domain averaged volume fractions ranging from 0.05 to 0.25. The operating conditions include unbounded sedimentation under gravity, as well as a turbulent-sluggish fluidization. In all cases, we discuss the influence of the meso-scale particle clusters on the macro-scale temperature distributions through the individual contributions of the unresolved terms in coarse-grid simulations. Thereby, we find that the dynamic estimation of the drift temperature leads to accurate results for the influence of local heat sinks on the global temperature difference decay between the phases and that the jump-response of a fluidized bed to an elevated gas-inflow temperature is correctly captured by the proposed multi-phase turbulence model. Furthermore, the importance of the correct estimation of the hydrodynamics for a correct prediction of the thermodynamics is highlighted.

A multiscale approach to hybrid RANS/LES wall modeling within a high‐order discontinuous Galerkin scheme using function enrichment

SummaryWe present a novel approach to hybrid Reynolds‐averaged Navier‐Stokes (RANS)/ large eddy simulation (LES) wall modeling based on function enrichment, which overcomes the common problem of the RANS‐LES transition and enables coarse meshes near the boundary. While the concept of function enrichment as an efficient discretization technique for turbulent boundary layers has been proposed in an earlier article by Krank & Wall (A new approach to wall modeling in LES of incompressible flow via function enrichment. J Comput Phys. 2016;316:94‐116), the contribution of this work is a rigorous derivation of a new multiscale turbulence modeling approach and a corresponding discontinuous Galerkin discretization scheme. In the near‐wall area, the Navier‐Stokes equations are explicitly solved for an LES and a RANS component in one single equation. This is done by providing the Galerkin method with an independent set of shape functions for each of these two methods; the standard high‐order polynomial basis resolves turbulent eddies, where the mesh is sufficiently fine and the enrichment automatically computes the ensemble‐averaged flow if the LES mesh is too coarse. As a result of the derivation, the RANS model is applied solely to the RANS degrees of freedom, which effectively prevents the typical issue of a log‐layer mismatch in attached boundary layers. As the full Navier‐Stokes equations are solved in the boundary layer, spatial refinement gradually yields wall‐resolved LES with exact boundary conditions. Numerical tests show the outstanding characteristics of the wall model regarding grid independence, superiority compared to equilibrium wall models in separated flows, and achieve a speed‐up by two orders of magnitude compared to wall‐resolved LES.

Variational Multiscale based dissipation models for the estimation of atmospheric seeing

In this work we present a numerical model for the estimation of atmospheric seeing in observation sites. The particularity of the method is that it is based on a Variational Multiscale turbulence model, its main feature being that the numerical mechanisms which are used to deal with stability issues (convection and the inf–sup condition for incompressible flows) do also take care of the modeling of turbulence. Based on this turbulence model, we develop the expressions for the viscous and thermal dissipations, ∊num and χnum, which are later used for evaluating the constant of structure of the refraction index Cn2 following the classical model developed by Tatarski. Numerical examples show the behavior of the proposed numerical scheme when applied to turbulent flow practical cases, which include a convective boundary layer, the flow inside a transfer optics room, and a telescope enclosure.

Numerical Investigation of Flow in an Asymmetric Plane Diffuser Using a Multi-Scale Turbulence Model

The flow in an asymmetric plane diffuser was simulated using both the multiscale turbulence model based on the variable interval time average method and the standardk-εmodel based on the Reynolds average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The capability of the multi-scale model to simulate flow in an asymmetric plane diffuser has been validated. Pressure coefficient, frictional resistance coefficient and mean velocity profiles downstream are in agreement with experiments. Moreover, the results predicted by the multi-scale model are better than that predicted by the standardk-εmodel. The computational results show that the multiscale turbulence model can successfully simulate this type of separated flow.

Numerical Investigation of Flow Past a Backward-Facing Step Using a Multi-Scale Turbulence Model

The flow past a backward-facing step was simulated using both the multiscale turbulence model based on the variable interval time average method and the standard k-ε model based on the Reynolds average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The capability of the multi-scale model to simulate flow past a backward-facing step has been validated. Both pressure coefficient and mean velocity profiles downstream are in agreement with experiments. Moreover, the results predicted by the multi-scale model are better than that predicted by the standard k-ε model. The computational results show that the multiscale turbulence model can successfully simulate this type of separated flow.

Numerical Investigation of the Gas-Liquid Two-Phase Flow around the Square-Section Cylinder Using a Multi-Scale Turbulence Model

The upward gas-liquid cross flow around a square cylinder was simulated using two fluid model with the multi-scale turbulent model based on the variable interval time average method. The computational results show that the multi-scale turbulent model can successfully simulate lift coefficient, drag coefficient and vortex shedding characteristics of flow around a body, and can also accurately predict the void fraction distribution and flow structure. Compared with the experimental data, the results of the multi-scale model are better than that of Standard k-ε model and RNG k-ε model. Hence, the study of this paper certificates further that this model can be used in the simulation of the gas-liquid flow around bluff bodies and outher engineering application.

Numerical Investigation of the Gas-Liquid Two-Phase Flow around the Square-Section Cylinder Using a Multi-Scale Turbulence Model

The upward gas-liquid cross flow around a square cylinder was simulated using two fluid model with the multi-scale turbulent model based on the variable interval time average method. The computational results show that the multi-scale turbulent model can successfully simulate lift coefficient, drag coefficient and vortex shedding characteristics of flow around a body, and can also accurately predict the void fraction distribution and flow structure. Compared with the experimental data, the results of the multi-scale model are better than that of Standard k-ε model and RNG k-ε model. Hence, the study of this paper certificates further that this model can be used in the simulation of the gas-liquid flow around bluff bodies and outher engineering application.

Numerical Investigation of Flow around a Triangle Cylinder Using a Multiscale Turbulence Model

The unsteady flow around a triangle cylinder was simulated using the multiscale turbulence model based on the variable interval time average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The computational results show that the multiscale turbulence model can successfully simulate vortex shedding characteristics. The Strouhal number and time-averaged velocity profiles also agree better with experiments than that of the standard k-ε model.

Numerical Investigation of Flow around a Triangle Cylinder Using a Multiscale Turbulence Model

The unsteady flow around a triangle cylinder was simulated using the multiscale turbulence model based on the variable interval time average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The computational results show that the multiscale turbulence model can successfully simulate vortex shedding characteristics. The Strouhal number and time-averaged velocity profiles also agree better with experiments than that of the standard k-ε model.

Numerical Investigation of Three-Dimensional Flow over a Swept Backward-Facing Step Using a Multi-Scale Turbulence Model

Numerical Investigation of Three-Dimensional Flow over a Swept Backward-Facing Step Using a Multi-Scale Turbulence Model

Numerical Investigation of Three-Dimensional Flow over a Swept Backward-Facing Step Using a Multi-Scale Turbulence Model

Using a multiscale model based on the variable interval time average method and a set of averaged equations for incompressible turbulent flows, we simulate the three-dimensional separated and reattaching flow over a swept backward-facing step. The results show that the reattachment length and vortex structure vary with the swept angles from 15° to 60° through a detailed study about velocity profile and pressure distribution. Generally the reattachment length and span wise vorticity decrease dramatically after =30°. The result fully accord with experimental data. Results show that this model is suitable for separated flows, and accurately predict the flow field, so this model should be useful in engineering application.

Residual-based variational multiscale turbulence models for unstructured tetrahedral meshes

This paper presents three-level residual-based turbulence models for the incompressible Navier–Stokes equations. Employing the variational multiscale (VMS) framework, the velocity and pressure fields are decomposed into two overlapping hierarchical scales, thereby leading to a system of coupled mixed field problems. The mixed problem at the fine scales is stabilized via a subsequent VMS application that results in a further decomposition of the fine-scale velocity field into level-I and level-II scales. The level-II scales are modeled using higher-order bubble functions that are then variationally embedded in the level-I formulation to stabilize it. The level-I problem is modeled via a second set of bubble functions that are linearly independent of the bubbles employed at level-II. Finally, the resulting level-I fine-scales are variationally embedded in the coarse-scale formulation. This yields a residual-based turbulence model for the larger or coarser-scales. A significant feature of the proposed method is that it results in a concurrent top-down and bottom up two-way nesting of the scales. In addition, the resulting turbulence model does not possess any embedded tunable parameters. Another attribute of the formulation is that the fine scales at every level are driven by the residuals of the Euler–Lagrange equations of the coarser scales at the preceding levels, thereby resulting in a method that is variationally consistent. Various algorithmic generalizations of the method are presented that lead to computationally economic residual-based turbulence models. The proposed telescopic depth in scales approach helps make these models accurate for low order tetrahedral and hexahedral elements, a feature that is facilitated by the higher-order bubble functions over element interiors and it results in an enhanced representation of the fine-scale terms modeling the fine viscous effects. From a computational perspective this method results in easy-to-implement equal-order pressure–velocity elements, and possesses the desirable p-refinement feature. Numerical performance of the method is assessed on turbulent channel flow problems at Re=395 and Re=590. Also presented are the results for turbulent SD-7003 airfoil at Re=60,000 and comparison is made with the published experimental data and numerical results.

Multiscale turbulence models based on convected fluid microstructure

The Euler-Poincaré approach to complex fluids is used to derive multiscale equations for computationally modeling Euler flows as a basis for modeling turbulence. The model is based on a kinematic sweeping ansatz (KSA) which assumes that the mean fluid flow serves as a Lagrangian frame of motion for the fluctuation dynamics. Thus, we regard the motion of a fluid parcel on the computationally resolvable length scales as a moving Lagrange coordinate for the fluctuating (zero-mean) motion of fluid parcels at the unresolved scales. Even in the simplest two-scale version on which we concentrate here, the contributions of the fluctuating motion under the KSA to the mean motion yields a system of equations that extends known results and appears to be suitable for modeling nonlinear backscatter (energy transfer from smaller to larger scales) in turbulence using multiscale methods.

A mathematical and physical Study of multiscale deconvolution models of turbulence

This work presents a rigorous analysis of mathematical and physical properties for solutions of multiscale deconvolution turbulence models. We show that solutions of these models exactly conserve model quantities for the integral invariants of fundamental physical importance: kinetic energy, helicity, and (in two dimensions) enstrophy. The kinetic energy conservation is the key that allows us to next apply the phenomenology of homogeneous, isotropic turbulence to establish the existence of a model energy cascade and, in particular, that the cascade exhibits enhanced energy dissipation in a secondary accelerated cascade, which ends at the model's microscale (which we establish is larger than the Kolmogorov microscale). We also prove that the model dissipates energy at the same rate as true turbulent flow, ∼ O(U3 ∕ L), independent of Reynolds number. Lastly, we prove the existence of global attractors for the model solutions; the proof of which also shows that solutions are actually one degree of regularity higher than previously known. Copyright © 2012 John Wiley & Sons, Ltd.

Multi-scale turbulence modeling and the maximum information principle

We discuss averaged turbulence modeling of multi-scales of length for an incompressible Newtonian fluid, with the help of the maximum information principle. We suppose that there exists a function basis to decompose the turbulent fluctuations in a flow of our concern into the components associated with various spatial scales and that there is a probability density function f of these fluctuation components. The unbiased form for f is determined and the turbulence model is closed, with the multi-scale correlations up to the fourth order, through maximizing the information under the constraints for that flow. Due to the computational difficulty to maximize the information, a closely related but simple alternative objective is sought, like the determinant or the trace of the second order correlations of the turbulent flow. The issues still yet to be resolved are indicated.

Stabilized space–time computation of wind-turbine rotor aerodynamics

We show how we use the Deforming-Spatial-Domain/Stabilized Space---Time (DSD/SST) formulation for accurate 3D computation of the aerodynamics of a wind-turbine rotor. As the test case, we use the NREL 5MW offshore baseline wind-turbine rotor. This class of computational problems are rather challenging, because they involve large Reynolds numbers and rotating turbulent flows, and computing the correct torque requires an accurate and meticulous numerical approach. We compute the problem with both the original version of the DSD/SST formulation and a recently introduced version with an advanced turbulence model. The DSD/SST formulation with the advanced turbulence model is a space---time version of the residual-based variational multiscale method. We compare our results to those reported recently, which were obtained with the residual-based variational multiscale Arbitrary Lagrangian---Eulerian method using NURBS for spatial discretization and which we take as the reference solution. While the original DSD/SST formulation yields torque values not far from the reference solution, the DSD/SST formulation with the variational multiscale turbulence model yields torque values very close to the reference solution.

Large eddy simulation of turbulent Taylor–Couette flow using isogeometric analysis and the residual-based variational multiscale method

We present an application of the residual-based variational multiscale turbulence modeling (RBVMS) methodology to the computation of turbulent Taylor–Couette flow at high Reynolds number. We show that the RBVMS formulation globally conserves angular momentum, a feature that is felt to be important for flows dominated by rotation, and that is not shared by standard stabilized formulations of fluid flow. Weak imposition of Dirichlet boundary conditions is employed to enhance the accuracy of the RBVMS framework in the presence of thin turbulent boundary layers near solid walls. Calculation of conservative boundary forces and torques is also presented for the case of weakly enforced boundary conditions. NURBS-based isogeometric analysis is employed for the spatial discretization, and mesh refinement is performed to assess the convergence characteristics of the proposed methodology. Numerical tests show that very accurate results are obtained on relatively coarse grids. To the best of the authors’ knowledge, this paper is the first to report large eddy simulation computations of this challenging test case.

Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes

In this work, we combine (i) NURBS-based isogeometric analysis, (ii) residual-driven turbulence modeling and iii) weak imposition of no-slip and no-penetration Dirichlet boundary conditions on unstretched meshes to compute wall-bounded turbulent flows. While the first two ingredients were shown to be successful for turbulence computations at medium-to-high Reynolds number [I. Akkerman, Y. Bazilevs, V. M. Calo, T. J. R. Hughes, S. Hulshoff, The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech. 41 (2008) 371–378; Y. Bazilevs, V.M. Calo, J.A. Cottrell, T.J.R. Hughes, A. Reali, G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197 (2007) 173–201], it is the weak imposition of no-slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher Reynolds number [Y. Bazilevs, T.J.R. Hughes. Weak imposition of Dirichlet boundary conditions in fluid mechanics, Comput. Fluids 36 (2007) 12–26; Y. Bazilevs, C. Michler, V.M. Calo, T.J.R. Hughes, Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4853–4862]. These three ingredients form a basis of a possible practical strategy for computing engineering flows, somewhere between RANS and LES in complexity. We demonstrate this by solving two challenging incompressible turbulent benchmark problems: channel flow at friction-velocity Reynolds number 2003 and flow in a planar asymmetric diffuser. We observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data. This lends some credence to the proposed approach, which we believe may become a viable engineering tool.

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.

Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows

We present an LES-type variational multiscale theory of turbulence. Our approach derives completely from the incompressible Navier–Stokes equations and does not employ any ad hoc devices, such as eddy viscosities. We tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows. In the calculations, we employed linear, quadratic and cubic NURBS. A dispersion analysis of simple model problems revealed NURBS elements to be superior to classical finite elements in approximating advective and diffusive processes, which play a significant role in turbulence computations. The numerical results are very good and confirm the viability of the theoretical framework.