Numerical Simulation Of Viscous Flow Research Articles

Quantum spin representation for the Navier-Stokes equation

We develop a quantum representation for Newtonian viscous fluid flows by establishing a mapping between the Navier-Stokes equation (NSE) and the Schrödinger-Pauli equation (SPE). The proposed nonlinear SPE incorporates the two-component wave function and the imaginary diffusion. Consequently, classical fluid flow can be interpreted as a non-Hermitian quantum spin system. Using the SPE-based numerical simulation of viscous flows, we demonstrate the quantum/wavelike behavior in flow dynamics. Furthermore, the SPE equivalent to the NSE can facilitate the quantum simulation of fluid dynamics. Published by the American Physical Society 2024

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.

Boundary conditions for fluid-dynamic parameters of a single-component gas flow with vibrational deactivation on a solid wall

Boundary conditions for fluid-dynamic parameters of a strongly non-equilibrium singlecomponent rarefied gas flow in the slip regime are obtained using kinetic-theory methods. The gas flow is described in the frame of the state-to-state approach assuming vibrational energy exchange as the slow relaxation process. The set of governing equations including conservation equations coupled with additional relaxation equations for vibrational state populations is presented. The gas-solid surface interaction is considered on the basis of the specular-diffusive model, and possible vibrational deactivation/excitation processes on the wall are taken into account. The obtained boundary conditions depend on the accommodation and deactivation coefficients along with the transport coefficients such as the multi-component vibrational energy diffusion and thermal diffusion coefficients; the thermal conductivity; the bulk and shear viscosity coefficients and the relaxation pressure. The dependence of boundary conditions on the normal mean stress has been obtained for the first time. In the particular case of the gas without internal degrees of freedom, the slip velocity and the temperature jump can be reduced to the well-known in the literature expressions. Implementation of the state-specific boundary conditions should not cause additional computational costs in numerical simulations of viscous flows in the state-to-state approach, since the slip/jump equations depend on the transport coefficients which have to be evaluated regardless of the boundary conditions used in the code.

Numerical and experimental assessment of nominal wake for a bulk carrier

In this paper, numerical and experimental assessment of nominal wake for a handymax bulk carrier at model and full-scale is performed. A circumferential averaged non-dimensional axial velocity distribution and integral value of nominal wake are obtained based on the measurements performed with the Pitot tube in the towing tank of Brodarski institute. Reynolds averaged Navier–Stokes equations along with three different turbulence models are utilized within double body simulations. The influence of the application of wall functions is investigated at model scale. Since the obtained differences between numerical solutions with and without wall functions are not large, full-scale simulations are performed with wall functions to avoid large cell number. The obtained numerical results are validated at model scale and satisfactory agreement is achieved. The significant scale effects are noticed in both integral value of nominal wake and circumferential averaged non-dimensional axial velocity distribution. Therefore, full-scale numerical simulations of viscous flow offer important benefits, especially in the optimization of the stern flow.

NUMERICAL SIMULATION OF THE VISCOUS FLOW AROUND A TANKER MODEL

In this paper, numerical simulation of viscous flow around a tanker model was carried out utilizing software package STAR-CCM+. A mathematical model based on Reynolds Averaged Navier-Stokes equations, a k-e turbulence model and Volume of Fluid method for describing the motion of two-phase media are given. Necessary boundary conditions for the mathematical model and the method of discretization are also described. The effect of grid density on the numerical results for the total resistance of the tanker model was investigated using three different grid densities. Two different types of the k-e turbulence model were implemented and deviations in the numerical results are highlighted. The results for the total resistance of the tanker model, obtained by numerical simulations, were validated against the experimental results. The experiments were performed in the towing tank of the Brodarski Institute for a wide range of Froude numbers. It was shown that for all three grid densities and for both types of the k-e turbulence model satisfactory agreement with the experimental results can be achieved for the whole range of Froude numbers. The scale effects were investigated by a Computational Fluid Dynamics study for the same tanker model in three different scales. Numerically calculated scale effects on wave resistance are reviewed.

SU2: the Open-Source Software for Non-ideal Compressible Flows

The capabilities of the open-source SU2 software suite for the numerical simulation of viscous flows over unstructured grid are extended to non-ideal compressible-fluid dynamics (NICFD). A built-in thermodynamic library is incorporated to account for the non-ideal thermodynamic characteristics of fluid flows evolving in the close proximity of the liquid-vapour saturation curve and critical point. The numerical methods, namely the Approximate Riemann Solvers (ARS), viscous fluxes and boundary conditions are generalised to non-ideal fluid properties. Quantities of interest for turbomachinery cascades, as loss coefficients and flow angles, can be automatically determined and used for design optimization.A variety of test cases are carried out to assess the performance of the solver. At first, numerical methods are verified against analytical solution of reference NICFD test cases, including steady shock reflection and unsteady shock tube. Then, non-ideal gas effects in planar nozzles and past turbine cascades, typically encountered in Organic Rankine Cycle applications, are investigated and debated. The obtained results demonstrate that SU2 is highly suited for the analysis and the automatic design of internal flow devices operating in the non-ideal compressible-fluid regime.

Numerical simulation of viscous flow past an oscillating square cylinder using a CIP-based model

The flow past an in-line forced oscillating square cylinder at Reynolds number of 200 is studied using an in-house code, named constrained interpolation profile method developed in Zhejiang University (CIP-ZJU). The model is established in the Cartesian coordinate system using the CIP method to discretise the Navier-Stokes equations. The fluid-structure interaction is treated as a multiphase flow of the liquid and solid phases to be solved simultaneously. An immersed boundary method is used to deal with the boundary of the solid body. The CFD model is first applied to the computation of the flow past a fixed square cylinder for its validation. Computations are then performed for the flow past a square cylinder oscillating in the streamwise direction. Considerable attention is paid to the symmetric and anti-symmetric modes of the vortex shedding in the oscillating square cylinder wake. Various oscillation amplitudes and frequencies are simulated and their effects on the vortex shedding modes are analyzed via Lissajous patterns of the unsteady lift coefficient. The relationship among the lift coefficient, the drag coefficient and the lock-on range is also investigated quantitatively.

Numerical simulations of viscous flow around the obliquely towed KVLCC2M model in deep and shallow water

By solving the unsteady Reynolds averaged Navier–Stokes (RANS) equations in combination with the k-ω SST turbulence model, the unsteady viscous flow around the obliquely towed tanker KVLCC2M model in both deep and shallow waters is simulated and the hydrodynamic forces, the surface pressure distribution, and the wake field are calculated. The overset grid technology is used to avoid the grid distortion in large drift angle cases. The effects of the free surface are taken into account. At the first stage, the deep water cases with five oblique angles are designed as the benchmark test cases. The predicted wake field, the surface pressure distribution and the hydrodynamic forces acting on the hull agree well with the corresponding experimental data, implying the capability of the present method in the prediction of the viscous flow around the tanker drifting in shallow water. A set of systematic computations with varying water depths and drift angles are then carried out to study the viscous flow around the model drifting in shallow water. The forces and moments, as well as the surface pressure distribution are predicted and analyzed. The most significant changes such as the increased stagnation pressure in the bow, the acceleration of the flow along the ship's sides and in the gap between ship and seabed, the lower hull pressure and finally, the stronger vortices along the bilges and weaker vortices with larger diameters in the wake are noticed.

Numerical simulation of viscous flow over a grooved surface by the lattice Boltzmann method

The motivation of this work is to investigate a grooved surface’s drag reduction. The viscous flow through a two-dimensional microchannel with the grooved surface is analyzed by the lattice Boltzmann method (LBM). The effects of the grooved surface on the streamline patterns, the velocity distributions near wall region and the fluid shear stress distributions on the walls at different Reynolds numbers are studied. In addition, the influences of the groove’s geometrical parameters on the grooved surface’s drag reduction are discussed. The numerical results confirm the grooved surface’s drag reduction and present the drag reduction law of the grooved surface.

Numerical investigation of submarine hydrodynamics and flow field in steady turn

This paper presents numerical simulations of viscous flow past a submarine model in steady turn by solving the Reynolds-Averaged Navier-Stokes Equations (RANSE) for incompressible, steady flows. The rotating coordinate system was adopted to deal with the rotation problem. The Coriolis force and centrifugal force due to the computation in a bodyfixed rotating frame of reference were treated explicitly and added to momentum equations as source terms. Furthermore, velocities of entrances were coded to give the correct magnitude and direction needed. Two turbulence closure models (TCMs), the RNG κ - e model with wall functions and curvature correction and the Shear Stress Transport (SST) κ - ω model without the use of wall functions, but with curvature correction and low-Re correction were introduced, respectively. Take DARPA SUBOFF model as the test case, a series of drift angle varying between 0° and 16° at a Reynolds number of 6.53×106 undergoing rotating arm test simulations were conducted. The computed forces and moment as a function of drift angle during the steady turn are mostly in close agreement with available experimental data. Though the difference between the pressure coefficients around the hull form was observed, they always show the same trend. It was demonstrated that using sufficiently fine grids and advanced turbulence models will lead to accurate prediction of the flow field as well as the forces and moments on the hull.

Numerical simulation on the motion characteristics of freely rising bubbles using smoothed particle hydrodynamics method

Based on the principle of virtual works, a multiphase smoothed particle hydrodynamics (SPH) model is further developed from the foundation of Hu X Y et al. (2006) and Grenier N et al. (2009). In the present model, the surface tension force implementation suitable for the multiphase flows with a large density ratio is applied, and this allows a good continuity at the multiphase interface. Artificial displacement correction is applied to keep the particles distributing uniformly in the whole flow field, and therefore any artificial viscous term is never needed; this is very important in the numerical simulation of viscous flows since the introduction of artificial viscosity changes the Reynolds number. Background pressure and interface sharpness force are added in the equation of state and the equation of momentum respectively to ensure the multiphase interface stability and smoothness; this is essential in the simulation of multiphase flows with large density difference at the multiphase interface. Two types of viscosity expressions suitable for multiphase flows are introduced and analyzed; the conclusion is that the formula proposed by Morris et al. (1997) and its similarly derived forms can give more accurate results. In the numerical validations, an oscillating droplet test is applied first to confirm the accuracy of the surface tension model and good results are achieved. This demonstrates that the artificial displacement and the interface sharp force will make negligible effects to the surface tension implementation. After that, two classic quantitative benchmarks of rising bubbles are simulated and the results of SPH agree well with the reference data. Moreover, in the two numerical benchmarks, the effect of the artificial displacement, the choice of the viscosity expression, and the type of the kernel function are compared and finally an optimal combination of these numerical aspects is recommended. Based on the above numerical investigations, the splitting process of an initially circular bubble is simulated and the numerical results agree well with the experimental data. In the last numerical case, the process of chasing and merging between two rising bubbles in vertical direction is simulated, based on which the mechanisms of these interesting interactions between two rising bubbles are analyzed. It is demonstrated in the present work that further improved multiphase SPH model may provide a potential method for the research of bubble dynamics.

Numerical simulation of viscous flow past a circular cylinder subject to a circular motion

This paper presents a computational study of the two-dimensional flow of a viscous incompressible fluid past a circular cylinder subject to a circular motion. We used the Fourier spectral analysis with finite difference approximations to integrate the two-dimensional unsteady Navier–Stokes equations which govern the motion. The numerical simulations reveal the occurrence of multiple lock-in regions. In addition, an increase and inflation in the values of the mean and RMS drag coefficients within these lock-in regions, is observed. Comparisons with previous numerical and experimental results verify the accuracy and the validity of the present study.

Microscopic numerical simulations of suspension with particle accumulation in porous media

Microscopic numerical simulations of viscous flow with fine particles suspended in three-dimensional periodic porous media are conducted using the solid–liquid two-phase lattice Boltzmann method. The diameter ratios of migratory particles to that of spherical frame sand grains are 0.16, 0.20 and 0.25 and the saturation of the fine particles ranges from 0.025 to 0.125. It is confirmed that, when the fine particles are fixed in space and do not move, friction determines the permeability and, therefore, their surface area should be accounted for. On the other hand, when they are moving, the permeability depends almost solely on their volume fraction. Taking both effects into account, a unified model for the reduction in apparent permeability caused by fine particles is proposed with combining conventional models of permeability of porous media and viscosity of suspension. In the model, the ratio of the number of trapped particles to that of mobile ones is a key factor.

A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis

This article contains part of the material of four introductory lectures given at the 12th school ``Mathematical Theory in Fluid Mechanics'', Spring 2011, at Kacov, Czech Republic, on ``Numerical simulation of viscous flow: discretization, optimization and stability analysis''. In the first lecture on ``Numerical computation of incompressible viscous flow'', we discuss the Galerkin finite element method for the discretization of the Navier-Stokes equations for modeling laminar flow. Particular emphasis is put on the aspects pressure stabilization and truncation to bounded domains. In the second lecture on ``Goal-oriented adaptivity'', we introduce the concept underlying the Dual Weighted Residual (DWR) method for goal-oriented residual-based adaptivity in solving the Navier-Stokes equations. This approach is presented for stationary as well as nonstationary situations. In the third lecture on ``Optimal flow control'', we discuss the use of the DWR method for adaptive discretization in flow control and model calibration. Finally, in the fourth lecture on ``Numerical stability analysis'', we consider the numerical stability analysis of stationary flows employing the concepts of linearized stability and pseudospectrum.

Numerical simulations of viscous flows around surface ship by level set method

Numerical simulations of viscous flows around surface ships by coupling the 3D incompressible RANS equations with level set method are presented in this paper. The finite difference method is used to discretize the RANS equations with turbulent model SST k-ω. The fully nonlinear boundary condition at the free surface is satisfied at each time step and the evolution of the free surface is achieved by using the level set method. The coupled solver is applied to a benchmark case of viscous flows around an advancing Wigley ship with various Froude numbers. The computational results are in excellent agreement with experimental data. The simulations reveal clearly the generation and evolution of bow and stern waves.

High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations

An approach for the creation of high-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations is proposed. New versions of up to the eighth order of accuracy inclusive are implemented. For smooth solutions, numerical experiments on a sequence of grids show that the approximate solutions produced by these versions converge to the exact one with a high order of accuracy as h → 0, where h is the maximal linear cell size of a grid. The numerical results obtained for the benchmark problem of the lid-driven cavity flow suggest that the collocations and least squares method is well suited for the numerical simulation of viscous flows.

Numerical simulation of viscous flow over non-smooth surfaces

The incompressible viscous flow over several non-smooth surfaces is simulated numerically by using the lattice Boltzmann method. A numerical strategy for dealing with a curved boundary with second-order accuracy for velocity field is presented. The drag evaluation is performed by the momentum-exchange method. The streamline contours are obtained over surfaces with different shapes, including circular concave, circular convex, triangular concave, triangular convex, regular sinusoidal wavy and irregular sinusoidal wavy, are obtained. The flow patterns are discussed in detail. The velocity profiles over different kinds of non-smooth surfaces are investigated. The numerical results show that the lattice Boltzmann method is reliable, accurate, easy to implement, and can provide valuable help for some engineering practices.

Numerical simulations of viscous flows using a meshless method

AbstractThis paper uses the element‐free Galerkin (EFG) method to simulate 2D, viscous, incompressible flows. The control equations are discretized with the standard Galerkin method in space and a fractional step finite element scheme in time. Regular background cells are used for the quadrature. Several classical fluid mechanics problems were analyzed including flow in a pipe, flow past a step and flow in a driven cavity. The flow field computed with the EFG method compared well with those calculated using the finite element method (FEM) and finite difference method. The simulations show that although EFG is more expensive computationally than FEM, it is capable of dealing with cases where the nodes are poorly distributed or even overlap with each other; hence, it may be used to resolve remeshing problems in direct numerical simulations. Flows around a cylinder for different Reynolds numbers are also simulated to study the flow patterns for various conditions and the drag and lift forces exerted by the fluid on the cylinder. These forces are calculated by integrating the pressure and shear forces over the cylinder surface. The results show how the drag and lift forces oscillate for high Reynolds numbers. The calculated Strouhal number agrees well with previous results. Copyright © 2008 John Wiley & Sons, Ltd.

Numerical Simulation of Viscous Flow in a Partially filled Co-rotating Twin Screw Extruder

Abstract The free-surface flow of fully melted material in the conveying section of a co-rotating twin screw extruder has been simulated numerically with the Finite Volume Method and an Eulerian multiphase model. In this model the interface is captured in an implicit manner with the help of the local volume fraction in a cell. The results presented here reveal the power and conveying characteristics of the twin screw configuration. Some interesting findings are discussed for the power characteristic in the fully filled state as well. The flow field and derived quantities are discussed with regards to the axial transport, symmetries in the flow field and locations of high energy dissipation. All results are compared to the fully filled state. Additionally, a new term, the volume flux based “conveying degree”, is introduced in this paper. The difference to the volume-based degree of fill is pointed out.

Numerical simulation of viscous flow around a system of bodies on the basis of the Navier-Stokes equations in stream function-vorticity variables

A method of numerical solution of the Navier-Stokes equations in stream function-vortex-pressure variables is suggested to be used in simulating a nonstationary, nonsymmetric flow of a laminar incompressible viscous stream around a pair of rotating cylinders in a channel. Results of calculations which illustrate application of the method under different conditions of flow around bodies are presented.