# Selected Publications

### M. Wasserman, Y. Mor-Yossef, J.B. Greenberg, A positivity-preserving, implicit defect-correction multigrid method for turbulent combustion, Journal of Computational Physics, Volume 316, July 2016

A novel, robust multigrid method for the simulation of turbulent and chemically reacting flows is developed. A survey of previous attempts at implementing multigrid for the problems at hand indicated extensive use of artificial stabilization to overcome numerical instability arising from non-linearity of turbulence and chemistry model source-terms, small-scale physics of combustion, and loss of positivity. These issues are addressed in the current work. The highly stiff Reynolds-averaged Navier-Stokes (RANS) equations, coupled with turbulence and finite-rate chemical kinetics models, are integrated in time using the unconditionally positive-convergent (UPC) implicit method. The scheme is successfully extended in this work for use with chemical kinetics models, in a fully-coupled multigrid (FC-MG) framework.

To tackle the degraded performance of multigrid methods for chemically reacting flows, two major modifications are introduced with respect to the basic, Full Approximation Storage (FAS) approach. First, a novel prolongation operator that is based on logarithmic variables is proposed to prevent loss of positivity due to coarse-grid corrections. Together with the extended UPC implicit scheme, the positivity-preserving prolongation operator guarantees unconditional positivity of turbulence quantities and species mass fractions throughout the multigrid cycle. Second, to improve the coarse-grid-correction obtained in localized regions of high chemical activity, a modified defect correction procedure is devised, and successfully applied for the first time to simulate turbulent, combusting flows.

The proposed modifications to the standard multigrid algorithm create a well-rounded and robust numerical method that provides accelerated convergence, while unconditionally preserving the positivity of model equation variables. Numerical simulations of various flows involving premixed combustion demonstrate that the proposed MG method increases the efficiency by a factor of up to eight (8) times with respect to an equivalent single-grid method, and by two (2) times with respect to an artificially-stabilized MG method.

### Y. Mor-Yossef, Robust turbulent flow simulations using a Reynolds-stress-transport model on unstructured grids, Computers & Fluids, Volume 129, April 2016

Progress toward a stable and efficient numerical treatment for the Reynolds-averaged Navier–Stokes equations with a Reynolds-stress-transport model on unstructured grids is presented. The unconditionally stable time marching scheme for Reynolds-stress-transport models, originally developed by the author for structured grids, is extended for unstructured grids using a finite volume method. The scheme guarantees the convergence of the fixed point iteration on the linearized problem. Moreover, the scheme is a positivity-preserving scheme, regardless of the time step. Thanks to the scheme characteristics, a spatially second-order discretization method for the Reynolds stress model equations (exactly as applied to the mean-flow equations) can be used, free of stability difficulties within the fixed point iterations. It is shown that the limiter has a dramatic influence on the convergence characteristics. Specifically, the limiter applied to the turbulence model variables was found to significantly influence the overall convergence behavior. Another key to the overall flow solver stability is a simple and robust procedure that is proposed to explicitly enforce all the realizability conditions of the Reynolds stress tensor. Two- and three-dimensional numerical flow simulations demonstrate the robustness of the overall flow solver for industrial applications.

### Y. Mor-Yossef, Unconditionally stable time marching scheme for Reynolds stress models, Journal of Computational Physics, Volume 276, November 2014

Progress toward a stable and efficient numerical treatment for the compressible Favre–Reynolds-averaged Navier–Stokes equations with a Reynolds-stress model (RSM) is presented. The mean-flow and the Reynolds stress model equations are discretized using finite differences on a curvilinear coordinates mesh. The convective flux is approximated by a third-order upwind biased MUSCL scheme. The diffusive flux is approximated using second-order central differencing, based on a full-viscous stencil. The novel time-marching approach relies on decoupled, implicit time integration, that is, the five mean-flow equations are solved separately from the seven Reynolds-stress closure equations. The key idea is the use of the unconditionally positive-convergent implicit scheme (UPC), originally developed for two-equation turbulence models. The extension of the UPC scheme for RSM guarantees the positivity of the normal Reynolds-stress components and the turbulence (specific) dissipation rate for any time step. Thanks to the UPC matrix-free structure and the decoupled approach, the resulting computational scheme is very efficient. Special care is dedicated to maintain the implicit operator compact, involving only nearest neighbor grid points, while fully supporting the larger discretized residual stencil. Results obtained from two- and three-dimensional numerical simulations demonstrate the significant progress achieved in this work toward optimally convergent solution of Reynolds stress models. Furthermore, the scheme is shown to be unconditionally stable and positive.

### D. Raveh, Y. Mor Yossef, Y. Levy, Flow Simulations for the First Aeroelastic Prediction Workshop Using the EZNSS Code, 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, January 2013

This paper presents numerical simulations that were performed with the EZNSS flow solver for the first NASA Langley Aeroelastic Prediction Workshop. Two configurations were studied, the Benchmark Supercritical Wing (BSCW) and the High Reynolds Number Aerostructural Dynamics (HIRENASD) model. The BSCW wing is a rigid wing that was studied at transonic flow conditions, at a fixed angle of attack. Static as well as time-accurate simulations were performed, using several computational meshes and turbulence models, with the purpose of predicting the pressure coefficient distribution at a wing section at 60% of the span, where pressure data was available from a wind tunnel experiment. All of the models predicted the shock location within 10% chords of its wind-tunnel location. None of the models predicted accurately the pressure recovery behind the shock on the upper and lower surfaces. While some turbulence models and computational setups resulted in a steady flow, some predicted flow unsteadiness, with fluctuations of the shock position and of the aerodynamic coef- ficient values. This may indicate that the case of the BSCW wing, at the studied flow conditions, is on the verge of buffet instability. The HIRENASD wing was studied for its elastic deformations and associated pressure coefficient distribution at three flow conditions. All of the studied flow conditions resulted in good correlation between the computed and experimental pressure coefficient data. Overall, it appears that the numerical simulations predicted well the transonic static aeroelastic response and the response to forced excitation in cases of attached flows. The transonic cases of detached flows behind a shock were found to be highly sensitive to the numerical parameters of the simulation, especially the turbulence model used.

### M. Wasserman, Y. Mor-Yossef, I. Yavneh, J.B. Greenberg, A robust implicit multigrid method for RANS equations with two-equation turbulence models, Journal of Computational Physics, Volume 229, Issue 16, August 2010

The design of a new, truly robust multigrid framework for the solution of steady-state Reynolds-Averaged Navier–Stokes (RANS) equations with two-equation turbulence models is presented. While the mean-flow equations and the turbulence model equations are advanced in time in a loosely-coupled manner, their multigrid cycling is strongly coupled (FC-MG). Thanks to the loosely-coupled approach, the unconditionally positive-convergent implicit time-integration scheme for two-equation turbulence models (UPC) is used. An improvement to the basic UPC scheme convergence characteristics is developed and its extension within the multigrid method is proposed. The resulting novel FC-MG-UPC algorithm is nearly free of artificial stabilizing techniques, leading to increased multigrid efficiency. To demonstrate the robustness of the proposed algorithm, it is applied to linear and non-linear two-equation turbulence models. Numerical experiments are conducted, simulating separated flow about the NACA4412 airfoil, transonic flow about the RAE2822 airfoil and internal flow through a plane asymmetric diffuser. Results obtained from numerical simulations demonstrate the strong consistency and case-independence of the method.

### Y. Kidron, Y. Mor-Yossef, and Y. Levy, Robust Cartesian Grid Flow Solver for High-Reynolds-Number Turbulent Flow Simulations, AIAA Journal, Vol. 48, No. 6 (2010)

A novel, Cartesian-grid-based flow solver is developed for predicting complex high-Reynolds-number turbulent flowfields. The Cartesian grid generator is based on the cut-cell approach using cell merge and Cartesian layer techniques. Cartesian layers imitate the structured grid approach in which the mesh is stretched gradually. Local refinement is added based on local surface curvature. As a turbulence closure model, the two equation k–!-TNT turbulence model is successfully implemented using an unconditionally positive-convergent implicit time integration scheme. The overall flow solver’s robustness and accuracy are verified using three challenging test cases. The numerical results convincingly demonstrate the robustness and accuracy of the flow solver, especially in predicting aerodynamic forces.

### Y. Mor-Yossef, Y. Levy, The unconditionally positive-convergent implicit time integration scheme for two-equation turbulence models: Revisited, Computers & Fluids, Volume 38, Issue 10, December 2009

The unconditionally positive-convergent implicit scheme for two-equation turbulence models, originally developed by Mor-Yossef and Levy, is revisited. A compact, simple, and uniform reformulation of the method for the use of both structured and unstructured grid based flow solvers is presented. An analytical proof of the scheme revision is given showing that positivity of the turbulence model solutions and convergence of the turbulence model equations are guaranteed for any time step. Numerical experiments are conducted, simulating two test cases of three-dimensional complex flow fields using structured and hybrid unstructured grids. To demonstrate the overall scheme’s robustness, it is applied to non-linear k-ω and non-linear k-ϵ turbulence models. Results from the numerical simulations show that the scheme exhibits very good convergence characteristics, is robust, and it always preserves the positivity of the turbulence model dependent variables, even for an infinite time step.

### Y. Moryossef, Y. Levy, Designing a positive second-order implicit time integration procedure for unsteady turbulent flows, Computer Methods in Applied Mechanics and Engineering, Volume 196, Issues 41–44, 1 September 2007

A positivity preserving implicit procedure for first order time integration of two-equation turbulence models is extended to second-order time integration utilizing dual-time stepping. Because the time derivative of the turbulence quantities may behave as a negative source term it is treated implicitly. The procedure is applied to a non-linear k – ϵ turbulence model using an unstructured flow solver. Numerical simulations of the unsteady flow about a square cylinder are conducted to demonstrate the positivity characteristics of the turbulence quantities and the robustness of the procedure and of the flow solver. Results from the numerical simulations favorably compare with results from experiments.

Keywords: Implicit time integration; Turbulence model; Positive; Dual-time; M-matrix; Unstructured grids

### Y. Moryossef, Y. Levy, Unconditionally positive implicit procedure for two-equation turbulence models: Application to k–ω turbulence models, Journal of Computational Physics, Volume 220, Issue 1, December 2006

A new general implicit procedure that guarantees the positivity of turbulence model equations dependent variables is presented. The implicit procedure is based on designing the implicit Jacobian to be an M-matrix. A unified approach for the treatment of the convection, diffusion and source terms is introduced and employed. An appropriate design of the M-matrix guarantees the positivity of the turbulence equation dependent variables for any time step, without the use of any clipping. The procedure is employed and demonstrated using a k–ω turbulence model. An efficient construction of the Jacobian matrix is proposed, resulting in diagonal matrices. The model is tested by simulating three test cases of two- and three-dimensional complex flow fields. Results from the numerical simulations show that the new procedure exhibits very good convergence characteristics, is extremely robust, and always preserves the positivity of the turbulence variables.

Keywords: Implicit method; Turbulence model; Positivity preservation; Numerical stiffness; Computations; Source term