Menter's Shear Stress Research Articles

Evaluating numerical models for the prediction of pollutant dispersion over Tokyo's Polytechnic University campus

This paper focuses on assessing how different Computational Fluid Dynamics (CFD) methodologies perform in modelling pollutant dispersion over extended urban areas. The case selected for evaluating the proposed methodologies is based on an extensive experimental study conducted by Tachibana et al. (2022), in which outdoor field measurements and wind tunnel experiments were carried out for monitoring tracer gas dispersion over the Atsugi Campus of Tokyo's Polytechnic University (TPU). Numerical results obtained by solving steady Reynolds Averaged Navier Stokes (RANS) governing equations, as well as using a scale resolving Large Eddy Simulation (LES) approach are compared against experimental data, in terms of both dimensionless local velocities and pollutant concentrations. The different meshing strategies that are employed for each methodology are also discussed in detail. The aim of the paper is to help gain further insights about the extent to which the Menter's Shear Stress Transport (SST) k-ω RANS model and LES can be used in the Architectural, Engineering and Construction (AEC) industry for pollutant dispersion modelling.

Mathematical modeling and performance evaluation of Ducted Horizontal-axis Helical Wind Turbines: Insights into aerodynamics and efficiency.

With the escalating demand for energy, there is a growing focus on decentralized, small-scale energy infrastructure. The success of new turbines in this context is notable. However, many of these turbines do not follow many of the basic ideas established to evaluate their performance, leaving no precise technique or mathematical model. This research developed a Ducted Horizontal-axis Helical Wind Turbine (DHAHWT). The DHAHWT is a duct-mounted helical savonius turbine with a venturi and diffuser to improve flow. Unlike a vertical axis helical savonius turbine, DHAHWT revolves roughly parallel to the wind, making it a horizontal turbine. This complicates mathematical and theoretical analysis. This study created a DHAHWT mathematical model. COMSOL simulations utilizing Menter's Shear Stress Transport model (SST) across an incoming velocity range of 1m/s to 4m/s were used to evaluate the turbine's interaction with the wind. MATLAB was used to train an artificial neural network (ANN) utilizing COMSOL data to obtain greater velocity data. The Mean Average Percentage Error (MAPE) and Root Mean Square Error (RMSE) of ANN data were found to be 3%, indicating high accuracy. Further, using advanced statistical methods the Pearson's correlation coefficient was calculated resulting in a better understanding of the relationship of between incoming velocity and velocity at different sections of the wind turbine. This study will shed light on the aerodynamics and working of DHAHWT.

A modification of production term for the shear stress transport turbulence model based on a new wall-law for adverse pressure gradient

This paper presents an attempt to improve the predictive accuracy of the Menter shear stress transport (SST) turbulence model for adverse pressure gradient (APG) flows. The foundation of this work lies in the newly modified law of the wall and turbulent kinetic energy (TKE) transport equation within the APG turbulent boundary layer. The modification of the law of the wall is achieved through a newly proposed variable λ, which is derived theoretically and calibrated using direct numerical simulation data. Based on the new law of the wall, we analyze the TKE transport equation of the SST model at APG and demonstrate that the analytical solution satisfying the law of the wall cannot achieve equilibrium in the current SST model. Further theoretical analysis reveals that this imbalance is caused by the TKE production term Pk, and the decomposition of the wall friction coefficient Cf indicates that Pk directly affects the prediction of flow separation. Based on the analysis and the new law of the wall, an additional term for Pk is suggested to correct Pk and improve the model's predictive accuracy for APG flows. Numerical validation results show that the correction leads to an increase in Pk within the APG turbulent boundary layer, resulting in more accurate predictions of Cf and the mean velocity profile in the inner layer of the turbulent boundary layer.

Data Assimilation of Ideally Expanded Supersonic Jet Using RANS Simulation for High-Resolution PIV Data

Data assimilation using particle image velocimetry (PIV) and Reynolds-averaged Navier–Stokes (RANS) simulation was performed for an ideally expanded supersonic jet flying at a Mach number of 2.0. The present study aims to efficiently reconstruct all the physical quantities in the aeroacoustic fields that match well with a realistic, experimentally obtained flow field. The two-dimensional, two-component PIV measurement was applied to the jet axis plane, and the time-averaged velocity field was obtained using single-pixel ensemble correlation. Two-dimensional axisymmetric RANS simulation using the Menter shear stress transport (SST) model was also performed, and the parameters of the SST model were optimized via data assimilation using the ensemble Kalman filter. The standard deviation of the observation noise σ, which is a parameter of the ensemble Kalman filter, is estimated by the previously proposed method (Nakamura et al., Low-Grid-Resolution-RANS-Based Data Assimilation of Time-Averaged Separated Flow Obtained by LES. Int. J. Comp. Fluid. Dyn., 2022), and its effectiveness was investigated for the first time. This method effectively estimated the magnitude of σ at each generation without tuning the hyperparameters. The assimilated flow fields exhibited similar flow structures observed in PIV such as the potential core length or shear layer. Therefore, the present framework can be used to estimate time-averaged full flow fields that match well with experimentally observed flow fields, and has the potential to construct a database for the Navier-Stokes-based stability analysis that requires a full flow field.

Improved SST turbulence model for supersonic flows with APG/separation

A goal in modeling the turbulent flows in computational fluid dynamics is to find an empirical yet universal model to address the complex scenarios. Menter's Shear Stress Transfer (SST) [1] is a Reynolds-Averaged Navier-Stokes (RANS) based two-equation turbulence model and has been one of the most widely used turbulence models for numerical simulation of viscous flows. However, in some practical cases on supersonic, the standard SST model shows unsatisfactory prediction of flow separation and the often underestimation of Reynolds stress. The error mainly comes from the inappropriate use of Bradshaw's assumption which is proposed under turbulent kinetic energy (TKE) equilibrium flow conditions. This paper proposes an improved SST turbulence model which combines the structural parameter with the ratio of the turbulence generation term to the dissipation term and imposes a mixing function to correct the logarithmic and wake parts of the region. Several supersonic test cases are carried out to verify the new model. The calculation results show that the new model exhibits better performance in capturing the separation zones than the baseline model (BSL) and the standard SST model.

Modeling of the flow and heat exchange in pipes with turbulators of viscous heat carriers in the laminar region, as well as in the transition to turbulent flow

Objective. Mathematical modeling of heat transfer in pipes with turbulators for viscous heat carriers at Reynolds numbers characteristic of laminar and transient flow regimes is carried out by the calculation method. The solution of the heat exchange problem for semicircular cross-section flow turbulators based on multiblock computing technologies based on the solution of the Reynolds equations (closed for the transient mode using the Menter shear stress transfer model) and the energy equation (on multi-scale intersecting structured grids) by the factorized finite-volume method (FCOM) was considered.Method. The calculation was carried out on the basis of a theoretical method based on the solution of the Reynolds equations, closed for transient modes using the Menter shear stress transfer model, and the energy equation on multiscale intersecting structured grids (FCOM), by a factorized finite-volume method.Result. Both local and averaged characteristics of the flow and heat exchange in pipes with turbulators for a viscous coolant for laminar and transient flow modes of the coolant were obtained using the FCOM method in the work, which made it possible to determine for these modes the levels of heat exchange intensification that satisfactorily correlate with the existing experiment.Conclusion. The calculated relative hydraulic resistance for low turbulators increases quite slightly, and for medium-altitude turbulators reaches 2÷2.5 to the critical Reynolds number, and subsequently it increases up to 3 times; for high turbulators, the relative hydraulic resistance increases up to 4 times even before the transition flow regime is reached, after which it increases up to 4.5 times. The calculated relative isothermal intensified heat exchange under the laminar flow regime of a viscous coolant for relatively high turbulators increases almost 2 times; for relatively medium heights of turbulators — almost one and a half, and for low relative heights, the intensification of heat exchange is insignificant.

Structural uncertainty quantification of Reynolds-Averaged Navier–Stokes closures for various shock-wave/boundary layer interaction flows

Accurate prediction of Shock-Wave/Boundary Layer Interaction (SWBLI) flows has been a persistent challenge for linear eddy viscosity models. A major limitation lies in the isotropic representation of the Reynolds stress, as assumed under the Boussinesq approximation. Recent studies have shown promise in improving the prediction capability for incompressible separation flows by perturbing the Reynolds-stress anisotropy tensor. However, it remains uncertain whether this approach is effective for SWBLI flows, which involve compressibility and discontinuity. To address this issue, this study systematically quantifies the structural uncertainty of the anisotropy for oblique SWBLI flows. The eigenspace perturbation method is applied to perturb the anisotropy tensor predicted by the Menter Shear–Stress Transport (SST) model and reveal the impacts of anisotropy on the prediction of quantities of interest, such as separation and reattachment positions, wall static pressure, skin friction, and heat flux. The results demonstrate the potential and reveal the challenges of eigenspace perturbation in improving the SST model. Furthermore, a detailed analysis of turbulent characteristics is performed to identify the source of uncertainty. The findings indicate that eigenspace perturbation primarily affects turbulent shear stress, while the prediction error of the SST model is more related to turbulent kinetic energy.

Turbulence Model Effects on Vortex Interaction Prediction on a Multiswept Delta Wing

In this study, vortical flows around a generic triple-delta wing geometry were investigated at transonic speeds. The focus of this work was the investigation of vortex interaction at medium to high angles of attack. To this end, numerical simulations were performed at two different Mach numbers, M=0.5 and M=0.85. To assess the performance of different turbulence models in predicting the complex flowfield, comparisons with experimental data were performed. In addition to the integral forces and moments, surface pressures from pressure-sensitive paint and velocity field data from particle image velocimetry were used. Across all investigated test cases, the best agreement with experimental results was achieved by the Menter shear stress transport turbulence model.

Mathematical Modelling of Heat Transfer in Pipes with Turbulators in the Laminar Region and in the Transition to Turbulent Flow

The article presents an analysis of mathematical modeling of heat transfer in pipes with turbulators at low Reynolds numbers characteristic of laminar and transient flow modes of heat carriers. The solution of the heat exchange problem for semicircular cross-section flow turbulators based on multi-block computing technologies based on the solution of Reynolds equations closed using the Menter shear stress transfer model and the energy equation (on multi-scale intersecting structured grids) by the factorized finite-volume method is considered. Both local and averaged characteristics of the flow and heat transfer to a pipe with a system of turbulators were obtained, which made it possible to determine the levels of heat exchange intensification for these regime ranges.

Investigation of Aerodynamic Support Interference for Testing a Semispan T Tail Aircraft Model

High-fidelity computational fluid dynamics (CFD) flow simulations were carried out to assess the sources of aerodynamic support interference of an aircraft half-model with a T-tail empennage mounted in a wind tunnel in an attempt to improve how well a semispan model can represent the corresponding full-span aircraft. The Bombardier research aircraft configuration (BRAC) model was used, which consists of a fuselage, a fuselage to wing fairing, a cruise wing with flap fairings, an engine nacelle and pylon, a vertical tail, and a horizontal stabilizer (HSTAB). The standard half-model support used at the National Research Council of Canada’s 1.5 m trisonic wind tunnel consists of a wall-mounted splitter plate to remove a sizable portion of the upstream tunnel wall boundary layer and a two-dimensional peniche sandwiched between the model symmetry plane and the splitter plate. CFD simulations of the BRAC model mounted in the wind tunnel were performed using the Menter shear stress transport turbulence model at a Mach number of 0.2. The CFD results for the standard (baseline) mounting arrangement compared favorably well with the half-model experimental data. The BRAC CFD model was then modified to provide reference “full-span” data and to examine several alternate model mounting arrangements. All simulations were compared with the ideal full-span reference simulations. Significant disturbances were found to be induced by the reflection plane boundary layer and primary horseshoe vortex on the flowfield surrounding the HSTAB, which had a direct impact on the measured overall model forces and moments.

Investigating Pressure Fluctuations on Marine Vessel Rudders: Numerical Results from Propeller–Rudder–Hull Interactions

Yuan, H.; Li, F.; Yan, Q.; Zhang, W., and Hu, J., 2023. Investigating pressure fluctuations on marine vessel rudders: Numerical results from propeller–rudder–hull interactions. Journal of Coastal Research, 39(2), 284–295. Charlotte (North Carolina), ISSN 0749-0208. Severe pressure fluctuations induced by propeller wake enhance rudder vibrations. In the present study, the pressure fluctuations on the rudder are numerically investigated by using k-ω Menter's shear stress transport turbulence model. The pressure fluctuations on the port and starboard sides of the rudder are compared in time and frequency domains. Furthermore, the impact of the advance coefficient on the displacement of propeller wake and pressure fluctuations is also discussed. The numerical results indicate that the propeller wake displacement is oriented upward and downward at the port and starboard sides, respectively. The most intense pressure fluctuation occurs at the propeller blade passage frequency. As a result of vortex displacement, the suction-side fluctuations are weaker than those on the pressure side at frequencies of 100–300 Hz over an advance coefficient range of J = 0.75–0.925. The displacement of the propeller wake weakens when the advance coefficient decreases.

Two-equation turbulent viscosity model for simulation of transitional flows: An efficient artificial neural network strategy

Laminar-to-turbulent transition phenomena are ubiquitous in natural and industrial flows. As to the Reynolds-averaged Navier–Stokes (RANS) simulation method, the workhorse for accurate prediction of such flow regime boils down to the consideration of the transition effect in turbulence modeling. In this paper, an industrial–practical transition–turbulence model with excellent accuracy, robustness, and efficiency is established by the fully connected artificial neural network (ANN), which maps the relation between the RANS mean flow variables and an intermittency factor. A one-equation local correlation-based transition model coupled with Menter's shear stress transport (SST) model is taken as the benchmark. The present two-way coupling ANN model is trained with two National Advisory Committee for Aeronautics (NACA) airfoils, that is, NACA0012 and NACA2418, at various angles of attack and Mach numbers, while tested with the A-airfoil, NACA0015, and RAE 2822 supercritical airfoils in different flow states. The a posteriori test results manifest that the mean pressure coefficient, skin friction coefficient, size of laminar separation bubble, mean streamwise velocity, Reynolds shear stress, and lift/drag/moment coefficient predicted by the ANN model are all in good agreement with those given by the benchmark transition-based SST model. Furthermore, the ANN model exhibits higher calculation efficiency and convergence speed than the traditional transition-predictive SST model. The present work may pave a new way for machine learning methods to be used in integrated transition–turbulence modeling toward industrial applications.

Transport-Based Transition Prediction for the Common Research Model Natural Laminar Flow Configuration

This paper presents the application of a new -based one-equation transition transport model to the Common Research Model Natural Laminar Flow Configuration (CRM-NLF), which was the central test case of the 1st AIAA CFD Transition Modeling and Prediction Workshop. The new model uses a simplified formulation of the Arnal, Habiballah, and Delcourt transition criterion that is stability based. The model is formulated in a grid-point-local manner and is coupled to the Menter shear stress transport eddy-viscosity turbulence model. The flow conditions applied are exactly those that were reported from the National Transonic Facility wind tunnel test campaign. The comparisons of the results yielded by the new model in terms of the transition fronts along the wing span show a very good match with the transition fronts derived from the temperature-sensitive paint images taken during the wind tunnel test and with computational result from an -based Two--factor strategy using critical -factors for Tollmien–Schlichting and crossflow instabilities and applying a local linear stability code.

Uncertainty analysis of turbulence model in capturing flows involving laminarization and retransition

Flows experiencing laminarization and retransition are universal and crucial in many engineering applications. The objective of this study is to conduct an uncertainty quantification and sensitivity analysis of turbulence model closure coefficients in capturing laminarization and retransition for a rapidly contracting channel flow. Specifically, two commonly used turbulence models are considered: the Spalart-Allmaras (SA) one-equation model and the Menter Shear Stress Transport (SST) two-equation model. Thereby, a series of steady Reynolds Averaged Navier-Stokes (RANS) predictions of aero-engine intake acceleration scenarios are carried out with the purposely designed turbulence model closure coefficients. As a result, both SA and SST models fail to capture the retransition phenomenon though they achieve pretty good performance in laminarization. Using the non-intrusive polynomial chaos method, solution uncertainties in velocity, pressure, and surface friction are quantified and analyzed, which reveals that the SST model possesses much great uncertainty in the non-laminar regime, especially for the logarithmic law prediction. Besides, a sensitivity analysis is performed to identify the critical contributors to the solution uncertainty, and then the correlations between the closure coefficients and the deviations of the outputs of interest are obtained via the linear regression method. The results indicate that the diffusion-related constants are the dominant uncertainty contributors for both SA and SST models. Furthermore, the remarkably strong correlation between the critical closure coefficients and the outputs might be a good guide to recalibrate and even optimize the commonly used turbulence models.

Modeling of heat transfer and hydraulic resistance in short and long straight round pipes with semicircular turbulators depending on geometric and operating parameters

Objective. The aim of the study was to determine the dependence of the distributions of average integral hydraulic resistances and convective heat transfer in turbulent flow in pipes with small (short channel) and large (long channel) sequences of semicircular periodic protrusions based on numerical solutions of the Reynolds equation systems, closed using Menter's shear stress transfer models, and energy equations on a multi-scale intersecting structured grid.Method. The calculation technique based on finite volume solutions of the Reynolds equation, closed using the Menter shear stress transfer model and the energy equation on a multi-scale intersecting structured grid, made it possible, with acceptable errors, to calculate the average coefficients of hydraulic resistance and heat transfer in a pipe with different quantities annular semicircular ledges.Result. Analytical comparisons were made of the calculated ratios for relative heat transfer and hydraulic resistance on the number of protrusions in channels with different values of relative heights of turbulators h/D, relative steps between turbulators t/D, different values of the Reynolds criteria Re, with other equivalent parameters, which showed that in in which case the qualitative deviations of the above characteristics are carried out monotonously, and in which case they are accompanied by extrema or inflections, and also showed cases of qualitative changes in the calculated characteristics. With the transition from 30 protrusions to 50, as a rule, only quantitative differences occur for the relative parameters of hydraulic resistance and heat transfer, and their qualitative changes are insignificant; with a further transition from 50 protrusions to 100, their quantitative changes also become insignificant.Conclusion. The nature of the patterns of distributions of the mean integral characteristics of flows and heat transfer for a channel with protrusions of various numbers must be taken into account for a short channel. An analysis of the calculated information obtained showed that when moving from a short channel with turbulators to a long one, most often, there is an increase in relative heat transfer and a decrease in relative hydraulic resistance, which justifies the advantage, from the point of view of intensification, of heat transfer by turbulence of the flow of the last channels in relation to the first channels.

The Effect of Curvature on the Heat Transfer Performance of Regenerative Cooling Passages for a High-Area-Ratio Nozzle

Abstract Numerical studies on the flow and heat transfer characteristics of rectangular regenerative cooling passages with lateral curvature for a high-area-ratio nozzle are presented. Unlike regular geometries of coolant paths used in various engineering applications, high-area-ratio rocket nozzles have steeply curved surfaces over which the cooling passages are provided. Though the inherent inward curvature benefits heat transfer enhancement in the throat region, it is insufficient to circumvent peak heat dissipation demand. Hence, the heat transfer enhancement due to lateral curvature of the coolant fluid passages provided near the throat region is explored in this study. Extensive numerical simulations have been performed to analyze the effect of the geometry of the cooling channel on its flow and heat transfer characteristics. The compressible turbulent flow field inside the nozzle has been resolved to understand the realistic local wall heat transfer characteristics of the typical high-area-ratio rocket nozzle using Advection Upstream Splitting Method (AUSM) scheme-based finite volume solver. Menter’s Shear Stress Transport k–ω turbulence model is used to model the turbulent flow inside the nozzle. Simulations of the incompressible coolant flow and conjugate heat transfer in regenerative cooling passages have been performed with realistic spatially varying local heat flux profiles, resulting due to compressible gas expansion in the convergent–divergent nozzle. Secondary flow structures are formed due to the lateral curvature of the coolant fluid passages and are found to enhance the heat transfer considerably. Further, the effect of coolant flowrate and channel curvature have been examined to explore its suitability to negotiate the peak heat flux dissipation demand at the throat region of the nozzle.

Iterated discretization error transport equations for laminar and turbulent flows

AbstractThe focus of this work is on discretization error estimation for finite‐volume schemes using iterated forms of the linear and nonlinear error transport equations. The accuracy of the discretization error estimates is studied by applying the error estimates as corrections to the primal problem and calculating the observed order of accuracy of the corrected solution. The test cases used in this work include two‐dimensional Cross Term Sinusoidal manufactured solutions for laminar flow and turbulent flow modeled by the Spalart–Allmaras and Menter's Shear Stress Transport model and a boundary layer manufactured solution for the Spalart–Allmaras model. The discretization error estimates of the linearized error transport equations and the nonlinear error transport equations are compared. Higher‐order accurate corrected solutions have been obtained which shows that the discretization error estimates are sufficiently accurate.

Suitability of helium gas as surrogate fuel for hydrogen in H2-Air non-reactive supersonic mixing studies

Owing to the safety concerns associated with the usage of hydrogen (H 2 ) fuel in conducting experiments, the current study investigates the suitability of using helium (He) gas as a surrogate fuel for H 2 for supersonic mixing studies. The study adopts a steady state computational approach to compare the mixing performance between the two injection cases, where a wall-based pylon-cavity aided flameholder configuration is used for the investigation. A common fuel injection location at the cavity floor is used for He and H 2 injection cases with an injector hole diameter of 1 mm. An inflow Mach number of 2.2 with sonic transverse fuel injection is kept the same for all the cases. A steady Reynolds-averaged Navier–Stokes (RANS) based computational model for compressed real gas is used for the investigation. The governing equations are solved by an implicit, second-order upwind solver with a two-equation Menter's Shear Stress Transport (SST) turbulence model. The study shows a variation in near field mixing performance between He and H 2 injection cases due to the difference in the molecular mass. As a result, a 10%–11% variation in the mixing efficiency is observed between the two cases. This makes He not suitable for micro-level mixing studies as a surrogate to H 2 . However, it is possible to closely predict the trend in global mixing performance parameters as in the H 2 injection case. • H2 injection shows an enhanced near field mixing than He injection. • Trend in global mixing performance between He and H2 injection are similar. • He cannot be used as a surrogate fuel for H2 in micro-level mixing studies.

Numerical Investigation of the Shallow Water Effect on the Total Resistance, Vertical Motion and Wave Profile of a Container Ship Model

A ship sailing in shallow water is affected by the interaction between the moving hull and the seabed in different forms such as: significant increase in total resistance, increase in the sinkage and trim that may result in squatting effect, a change in wave pattern and increased wave amplitude, a change in the propeller wake field and an altered propeller and manoeuvring performance. In order to investigate the influence of shallow water on a container ship that is assumed to be subjected to work in different water depths, the KRISO container ship model is analysed in both deep and shallow water, with a special focus on the change in resistance, vertical motion and wave profile. A viscous flow simulation is performed first to predict the ship performance in deep water and the results are compared with the experimental data that are available in the public domain. Then, the critical shallow water condition is investigated and the obtained results are compared against those formerly obtained in deep water. The numerical simulations are performed using the viscous flow solver ISIS-CFD of the FINETM/Marine software provided by NUMECA. The solver is based on the finite volume method to build the spatial discretization of the transport equation in order to solve the Reynolds-Averaged Navier-Stokes (RANS) equations. Closure to turbulence is achieved using the Menter Shear Stress Transport (K-ω SST) model, while the free-surface is captured through an air-water interface based on the Volume of Fluid (VOF) method. The comparison between the numerically obtained results and the available EFD data showed a satisfactory congruence.

Multifidelity modeling similarity conditions for airfoil dynamic stall prediction with manifold mapping

PurposeThe purpose of this work is to investigate the similarity requirements for the application of multifidelity modeling (MFM) for the prediction of airfoil dynamic stall using computational fluid dynamics (CFD) simulations.Design/methodology/approachDynamic stall is modeled using the unsteady Reynolds-averaged Navier–Stokes equations and Menter's shear stress transport turbulence model. Multifidelity models are created by varying the spatial and temporal discretizations. The effectiveness of the MFM method depends on the similarity between the high- (HF) and low-fidelity (LF) models. Their similarity is tested by computing the prediction error with respect to the HF model evaluations. The proposed approach is demonstrated on three airfoil shapes under deep dynamic stall at a Mach number 0.1 and Reynolds number 135,000.FindingsThe results show that varying the trust-region (TR) radius (λ) significantly affects the prediction accuracy of the MFM. The HF and LF simulation models hold similarity within small (λ ≤ 0.12) to medium (0.12 ≤ λ ≤ 0.23) TR radii producing a prediction error less than 5%, whereas for large TR radii (0.23 ≤ λ ≤ 0.41), the similarity is strongly affected by the time discretization and minimally by the spatial discretization.Originality/valueThe findings of this work present new knowledge for the construction of accurate MFMs for dynamic stall performance prediction using LF model spatial- and temporal discretization setup and the TR radius size. The approach used in this work is general and can be used for other unsteady applications involving CFD-based MFM and optimization.