Solution Of Viscous Flow Research Articles

General Solutions to the Navier-Stokes Equations for Incompressible Flow

<i>Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.</i>

[formula omitted]-adaptive discontinuous Galerkin solution of transonic viscous flows with variable time step-size

The discontinuous Galerkin methods demonstrated to be well suited for scale resolving simulations of complex configurations, characterized by different fluid dynamics phenomena, such as transition, separation, shock/boundary layer interaction. Unfortunately, solvers based on these methods cannot yet reach the computational efficiency of well-established standard solvers, e.g., based on finite volume methods. To reduce the computational cost and the memory footprint, while not spoiling the accuracy, different approaches were investigated, which act on the spatial or the temporal discretization, and on the linear algebra. The above approaches have been extensively investigated, but rarely considering their mutual interactions. In this work, a p-adaptation algorithm is coupled with a time step-size adaptation algorithm to mitigate robustness issues arising after each adaptation cycle, probably related to the projection of the old solution on the new polynomial orders distribution. Moreover, this strategy is able to control the transient phase before each adaptation cycle, and can handle also the presence of initial steady solution, when the mesh is too coarse and the polynomial degree has not yet compensated for the lack of spatial accuracy. The effectiveness and accuracy of the proposed algorithm are demonstrated on transonic viscous flows with shock-wave/boundary-layer interaction.

Global solutions for two-dimensional viscous pressureless flows with large variations of density

Global solutions for two-dimensional viscous pressureless flows with large variations of density

Numerical viscosity control in Godunov-like smoothed particle hydrodynamics for realistic flows modeling

In this study, we introduce a way to control the viscosity of the numerical approximation in the Godunov-like smoothed particle hydrodynamics (SPH) methods. This group of SPH methods includes momentum and energy fluxes in the right-hand sides of the equations, which are calculated by the solution of the Riemann problem between each pair of neighboring particles within the support radius of the smoothing kernel, which is similar to the procedure for the calculation of fluxes across cell boundaries in Godunov schemes. Such SPH methods do not require the use of artificial viscosity since the significant numerical viscosity is already introduced by a Riemann problem solution. We demonstrate that such a numerical viscosity may be measured and obtain the explicit expression for it depending on smoothed particle properties. In particular, we have found that Godunov-like SPH method with interparticle contact algorithms produces numerical viscosity several orders of magnitude higher than physical viscosity in materials. Modern approaches, such as SPH with monotonic upstream-centered scheme for conservation laws or weighted essentially non-oscillatory reconstruction techniques, have not only lower numerical viscosity but also too large for modeling real-world viscous flows. By constructing a correcting viscous stress tensor based on the analytical solution for discontinuous viscous flow, it is possible to reduce the viscous stresses of numerical origin. The use of such a correction makes it possible to improve the agreement with experiments in the simulation of viscous flows without using schemes of higher order reconstruction.

On the global-in-time inviscid limit of the 3D degenerate compressible Navier-Stokes equations

In this paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power ((ρϵ)δ with δ>1), for regular solutions to the corresponding Cauchy problem, via introducing one “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the velocity near the vacuum, we establish the uniform energy estimates for the local sound speed in H3 and (ρϵ)δ−12 in H2 with respect to the viscosity coefficients for arbitrarily large time under some smallness assumption on the initial density. Second, by making full use of this structure's quasi-symmetric property and the weak smooth effect on solutions, we prove the strong convergence of the regular solutions of the degenerate viscous flow to that of the inviscid flow with vacuum in H2 for arbitrarily large time. It is worth pointing out that the result obtained here seems to be the first one on the global-in-time inviscid limit of solutions with large velocities and vacuum for compressible flow in 3D space without any symmetric assumption.

Numerical Solution of Inviscid and Viscous Flow Across NACA0010 Airfoil with different Angles of Attack

The role of the airfoil in all engineering fields is very important. Airfoil designs are utilized in various applications such as aircraft wings, helicopter rotor blades, sails, fans, and turbine compressors. In the case of aircraft, airfoils are accountable for generating lift force and determining if it is adequate for balancing the aircraft's weight, as well as determining the amount of drag force that should be applied to the body. They also play a vital role in optimizing blade performance, as well as the heat and operational efficiency of turbines. Airfoils are moving in the flow passing by them, and they show different responses based on many parameters such as whether the flow is smooth or turbulent, viscous or inviscid, flow velocity, angle of attack, multidimensionality of velocity components, etc. For aeronautics vehicles and wind turbines, a large number of studies have been conducted to analyze aerodynamic forces on two dimensional airfoils as well as flow phenomena associated with different Reynolds numbers. Main object of this research is to figure out the flow characteristics around the airfoil body and analyze aerodynamic performance of airfoil. In order to investigate the computational efficiency of flow models, numerical examination has been done for flow across NACA0010 airfoil by using 2D Computational Fluid Dynamics (CFD) solver, ANSYS FLUENT 18.0, at various angle of attack (0o to 13o) for inviscid flow and (-30o to 30o) for viscous flow. Lift and drag coefficients and C p for lower and upper airfoil surface and velocity diagram based on different angle of attack has been solved for the sample under two different flow regimes. The contours of velocity were simulated, and stall point was noticed from 13° angle of attack in inviscid flow and 24o in viscous flow. In the non-viscous state, lift coefficient and drag coefficient was higher, in comparison with viscous flow.

Improved perturbation solution for viscous flow in a dilating–contracting permeable channel with velocity slip

Recently, Mandal and Ghosh [“Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip,” Phys. Fluids 35, 047121 (2023)] constructed perturbation solutions for viscous flow in porous channels with a slip condition and moving walls restricted to slow wall dilation–contraction rates. Herein, we show that this “slowness” assumption may be completely removed. In doing so, we develop a more widely applicable and more accurate perturbation scheme for all dilation–contraction rates. Our strategy involves generating new and exact solutions to the linear, inviscid problem with slip condition, and then we draw on this precise form to construct more accurate perturbation expansions for solutions to the nonlinear flow model than are currently available.

Improved perturbation solution for two-dimensional viscous flow between expanding or contracting permeable walls

Despite the continuing interest in the transport of biological fluid within contracting or expanding vessels, our knowledge is yet to be fully developed, even in the two-dimensional case. For example, explicit solutions and close approximations to these models remain unknown, and the physical problem has been restricted to the “slow” expansion or contraction of the walls. Thus, the purpose of this short communication is to partially address such challenges and gaps by generating explicit solutions and improving approximations to the flow problem without the “slowness” restriction being imposed. We show that when the Reynolds number is zero (i.e., the inviscid case), the corresponding homogeneous differential equation under consideration may be completely solved. We then illustrate how this exact solution may be leveraged to form more precise approximations to the flow via perturbation techniques when the Reynolds number is small. Our perturbation approach is only in one parameter (the Reynolds number) instead of the usual two parameters (the Reynolds number and wall dilation rate), and thus we make no restriction regarding the “slowness” of wall expansion or contraction for our general perturbation scheme. Our act of “shining new light through old windows” improves and extends the results of Majdalani, Zhou and Dawson and, moreover, our method has significant potential to be applied by researchers to form more precise one-parameter perturbation approximations to flow problems in contrast to the limitations of the traditional two-parameter perturbation approaches that have dominated the literature.

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Inviscid limit of compressible viscoelastic equations with the no-slip boundary condition

The inviscid limit for the two-dimensional compressible viscoelastic equations in the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions due to the appearance of strong boundary layers. However, when the deformation tensor effect is taken into account, our results show that the deformation tensor plays an important role in the vanishing viscosity process and can surprisingly prevent the formation of strong boundary layers. As a result we are able to justify the inviscid limit of solutions for the compressible viscous flows under the no-slip boundary condition governed by the viscoelastic equations, based on the uniform conormal regularity estimates achieved in this paper.

Multiple series solutions of viscous flows over a stretching/shrinking and porous sheet

The viscous fluid-flow over a stretching (shrinking) and porous sheets of non-uniform thickness is investigated in this paper. The modeled problem is presented by utilizing the stretching/shrinking and porous velocities and variable thickness of the sheet. Consequently, the new problem reproduces the different available forms of flow motion maintained over a stretching/shrinking and porous sheet of variable thickness in one go. As a result, the governing equations are embedded in several parameters which can be transformed into classical cases of stretched/shrunk flows over porous sheets. A set of general, unusual and new variables is formed in order to simplify the governing PDE and boundary conditions. Three different series solutions of the final ODE are presented. A single analytical solution is not sufficient to predict the exact effects of all parameters on the flow field properties. The problem is solved by a power and two asymptotic series methods. The results are verified by providing a powerful numerical solution the problem. A complete set of solutions is provided and comparison of the solutions with classical models is established for appropriate values of the parameters which is shown in different graphs and tables.

Successive Complementary Expansion Method for Solving Supersonic Laminar Flow Over Thin Airfoils& Numerical Simulation

The aim of the present study is two-fold. Firstly, it attempts to solve the asymptotic model of the 2D stationary potential flow of a supersonic viscous fluid over thin airfoils. In the case of a high Re number: Re=O(10SUP5/SUP), and a free stream Mach number of MSUB∞/SUB = 1.45, the asymptotic solution is built using the new successive complementary expansion method. The theoretical Mach number of the flow is estimated in the outer inviscid zone along the airfoil surface, and a correction term is added to account for the viscous boundary layer effect in order to get an accurate approximation throughout the whole domain.BR Secondly, for comparison purposes, numerical simulations of the NACA 43013 airfoil were run for the Mach number distribution in each area. On the one hand, when compared to numerical simulations, the asymptotic supersonic laminar viscous flow solution shows good agreement for the laminar boundary layer but begins to diverge slightly from 𝒙 = 𝟎, 𝟑.At this point, the laminar-turbulent transition starts, then the turbulent boundary layer develops towards the trailing edge. On the other hand, the Mach number distribution for the outer inviscid region matches the simulation and illustrates the change in velocity upon interaction with both the detached shock wave and the expansion waves.

Dual Solution with Heat Transfer through Moving Porous Plate of an Unsteady Incompressible Viscous Fluid

We examine graphically the behavior of the dual solution of an unsteady incompressible viscous flow with heat transfer due to a moving porous surface. The mathematical model is developed with the implementation of conservation laws which are then tackled numerically, and the simulations are performed through the Keller–Box technique. Application of the Keller–Box technique allows the non-linear differential equations to transform to a scheme of linear differential equations which are easier to illuminate. Outcomes of velocity and thermal behavior are shown graphically for certain values of the dimensionless parameters. It is obvious from the results that temperature of fluid decreases with the increasing values of dimensionless Prandtl number Pr, whereas the velocity enhances for blowing situations.

A meshless solution of the compressible viscous flow in axisymmetric tubes with varying cross-sections

The meshless Diffuse Approximate Method (DAM) is for the first time formulated and applied to simulate the compressible Newtonian flow of an ideal gas in an axisymmetric tube with varying cross-sections. The problem is structured by coupled partial differential equations for conservation of mass, momentum and energy, and the equation of state closure. These equations are solved in primitive variables and strong form. DAM is formulated on irregular node arrangement by using the second and third-order polynomial shape functions and Gaussian weights, leading to weighted least squares approximation on overlapping local subdomains. Pressure-velocity coupling is performed by the Pressure Implicit with Splitting of Operators (PISO) scheme. The solution of the represented novel application of DAM is verified by matching the meshless results with the fine-mesh finite-volume method results. The characteristics of meshless DAM for this kind of problem are systematically assessed by a detailed investigation of the varying node density, shape function order, Gaussian weight's shape, and the number of nodes in a local subdomain. The sensitivity study of DAM parameters shows that for the tested problems, the most suitable values of the Gaussian weight function and the number of nodes in a local subdomain are 5.0 and 13, respectively. Third-order convergence rate with better results is observed while using third-order polynomial shape functions.

A vertex-based reconstruction for cell-centered finite-volume discretization on unstructured grids

Recently, a vertex-based spatial reconstruction for unstructured cell-centered finite volume method has been proposed, showing advantages in accuracy, convergence, and efficiency. However, this method was only applied for inviscid flows. Moreover, for shock-capturing computations, the conventional cell-based limiter was the only choice to suppress numerical oscillations. In this work, the vertex-based reconstruction is extended for the solution of viscous flows, while automatically avoiding the typical decoupling problem. Moreover, a WENO-type nonlinear weighting strategy is designed and combined with the vertex-based reconstruction method, resulting in a simple and cost-efficient alternative to conventional slope limiters. In addition, the present method is combined with the recently proposed iterative near-boundary treatment, ensuring linear exactness even in the vicinity of boundaries, without sacrificing the overall computational efficiency. A series of numerical test cases involving viscous flows and shock waves provide evidence of the superior performance of the present method. The advantages of the present method are especially prominent on high aspect-ratio irregular triangular grids, which is a beneficial property for solving realistic fluid dynamic problems at high Reynolds numbers.

Stationary solutions of incompressible viscous flow in a wall-driven semi-circular cavity

Stationary numerical solutions of incompressible viscous flow inside a wall-driven semicircular cavity are presented. After a conformal mapping of the geometry, using a body-fitted mesh, the Navier-Stokes equations are solved numerically. The stationary solutions of the flow in a wall-driven semi-circular cavity are computed up to Re = 24000. The present results are in good agreement with the published results found in the literature. Our results show that as the Reynolds number increases, the sizes of the secondary and tertiary vortices increase, whereas the size of the primary vortex decreases. At large Reynolds numbers, the vorticity at the primary vortex centre increases almost linearly stating that Batchelor’s mean-square law is not valid for wall-driven semi-circular cavity flow. Detailed results are presented and also tabulated for future references and benchmark purposes.

Temporal stability of multiple similarity solutions for porous channel flows with expanding or contracting walls

In this paper, the temporal stability of multiple similarity solutions (flow patterns) for the incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is analyzed. This work extends the recent results of similarity perturbations of Sun et al. [“Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls,” Appl. Math. Modell. 77, 738–755 (2020)] by examining the temporal stability with perturbations of general form (including similarity and nonsimilarity forms). Based on the linear stability theory, two-dimensional eigenvalue problems associated with the flow equations are formulated and numerically solved by a finite difference method on the staggered grids. The linear stability analysis reveals that the stability of the solutions is same with that under the perturbations of a similarity form within the range of the wall expansion ratio α. [−5≤α≤3 as in Sun et al., “Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls,” Appl. Math. Modell. 77, 738–755 (2020)]. Further, it is found that the expansion ratio α has a great influence on the stability of type I flows: in the case of wall contraction (α<0), the stability region of the cross-flow Reynolds number (R) increases as the contraction ratio (|α|) increases; in the case of wall expansion and 0<α≤1, the stability region increases as the expansion ratio (α) increases; in the case of 1≤α≤3, type I flows are stable for all R where they exist. The flows of other types (types II and III with −5≤α≤3 and type IV with α = 3) are always unstable. As a nonlinear stability analysis or a validation of the linear stability analysis, the original nonlinear two-dimensional time-dependent problem with an initial perturbation of general form over those flow patterns is solved directly. It is found that the stability with the nonlinear analysis is consistent with the linear stability analysis.

Semi-Analytical Solution of Two-Dimensional Viscous Flow through Expanding/Contracting Gaps with Permeable Walls

In this research, the analytical methods of the differential transform method (DTM), homotopy asymptotic method (HAM), optimal homotopy asymptotic method (OHAM), Adomian decomposition method (ADM), variation iteration method (VIM) and reproducing kernel Hilbert space method (RKHSM), and the numerical method of the finite difference method (FDM) for (analytical-numerical) simulation of 2D viscous flow along expanding/contracting channels with permeable borders are carried out. The solutions for analytical method are obtained in series form (and the series are convergent), while for the numerical method the solution is obtained taking into account approximation techniques of second-order accuracy. The OHAM and HAM provide an appropriate method for controlling the convergence of the discretization series and adjusting convergence domains, despite having a problem for large sizes of obtained results in series form; for instance, the size of the series solution for the DTM is very small for the same order of accuracy. It is hard to judge which method is the best and all of them have their advantages and disadvantages. For instance, applying the DTM to BVPs is difficult; however, solving BVPs with the HAM, OHAM and VIM is simple and straightforward. The extracted solutions, in comparison with the computational solutions (shooting procedure combined with a Runge–Kutta fourth-order scheme, finite difference method), demonstrate remarkable accuracy. Finally, CPU time, average error and residual error for different cases are presented in tables and figures.

Investigation on shear layer instabilities and generation of vortices during shock wave and boundary layer interaction

The interaction of a reflected shock with the boundary layer inside the shock tubes is an important engineering problem. Studies related to shock wave mitigation and attenuation are performed inside the shock tubes. A proper understanding of the flow behind the reflected shock and the separated zone involving multiple vortical structures is highly essential for estimating the effectiveness of the shock wave mitigation/ attenuation. Such complex flows consist of Lambda shock, shear layer originating from the triple point, multiple shocklets, Mach stems, and vortices. Experimentally the shock structures are obtained through optical techniques. The vortices present in the compressible flow can be obtained through numerical simulations. The complex flows consisting of the above- mentioned features have been simulated numerically so far for the Reynolds numbers up to 1000 [ Zhou G, Xu K, Liu F. Grid-converged solution and analysis of the unsteady viscous flow in a two-dimensional shock tube. Phys Fluids 2018;30(016102):1-21.]. In the present investigation, the shock-wave boundary-layer interaction is simulated for the Reynolds numbers of 1000 and 2500 using a 13th order hybrid scheme to discern the distinct flow features. First, the solver [Kundu A, De S. Navier-Stokes simulation of shock-heavy bubble interaction: Comparison of upwind and WENO schemes. Comput Fluids 2017;157:131-145.] was validated with the benchmark wall density data for a Reynolds number of 1000. Next, the simulations were performed using 50 and 109.5 million cells for the Reynolds number of 2500. The density gradients, vorticity, wall density, and Fourier spectra were used for comparing the flow field for the Reynolds numbers of interest. The Lambda shock, Kelvin-Helmholtz (K-H) vortices in the shear layer, shocklets, the height of lambda shock, and Mach stems were obtained using a grid-mesh of 109.5 million cells. It is observed that the number of vortices generated inside the separated flow region increased with the increase in Reynolds number from 1000 to 2500. Furthermore, the triple point height and the number of K-H vortices generated at the shear layer also increase with an increase in Reynolds number. The present simulations revealed the formation of vortices close to the wall at a Reynolds number of 2500. Such flow structures have an important role in shock / blast wave mitigation and the associated aeroacoustics.

Exact solutions for quasi-one-dimensional compressible viscous flows in conical nozzles

Abstract