Unsteady Flow Problems Research Articles

A simplified axisymmetric lattice Boltzmann method for incompressible swirling and rotating flows

A simplified axisymmetric lattice Boltzmann method (SALBM) is developed in this paper for effective simulation of incompressible swirling and rotating flows. This model explores an alternative approach of reconstructing an axisymmetric model within the lattice Boltzmann (LB) framework. Essentially, SALBM reconstructs solutions to the macroscopic governing equations recovered from the axisymmetric LB equation through the Chapman-Enskog expansion analysis. Two variations of schemes, which bear different orders of temporal accuracy and are, respectively, suitable for the steady and the unsteady axisymmetric flow problems, can be evolved from SALBM. The proposed schemes reflect direct evolution of macroscopic variables instead of distribution functions, which could reduce the cost in virtual memory. Meanwhile, analytical interpretation of physical boundary conditions is available in SALBM which avoids tedious transformations as required in conventional LB models. Numerical tests further reveal that SALBM performs better than the existing axisymmetric LB models in numerical stability. These merits endow the present SALBM with advantages over previous models and forge its prospect in engineering applications.

A New Approximate Analytical Solutions for Two- and Three-Dimensional Unsteady Viscous Incompressible Flows by Using the Kinetically Reduced Local Navier-Stokes Equations

In this work, the kinetically reduced local Navier-Stokes equations are applied to the simulation of two- and three-dimensional unsteady viscous incompressible flow problems. The reduced differential transform method is used to find the new approximate analytical solutions of these flow problems. The new technique has been tested by using four selected multidimensional unsteady flow problems: two- and three-dimensional Taylor decaying vortices flow, Kovasznay flow, and three-dimensional Beltrami flow. The convergence analysis was discussed for this approach. The numerical results obtained by this approach are compared with other results that are available in previous works. Our results show that this method is efficient to provide new approximate analytic solutions. Moreover, we found that it has highly precise solutions with good convergence, less time consuming, being easily implemented for high Reynolds numbers, and low Mach numbers.

Numerical Analysis of Heat and Mass Transfer Flow of Nanofluid over a Moving Wedge Using Spectral Quasilinearization Method

In this paper the problem of unsteady two-dimensional heat and mass transfer flow of nanofluid past a moving wedge is considered. The effects of nanoparticle volume fraction, viscous dissipation, chemical reaction, and convective boundary conditions are studied. The physical problem is modeled using partial differential equations. Using suitable similarity variables, the governing equations and their related boundary conditions are transformed into dimensionless forms of a system of coupled nonlinear ordinary differential equations. The resulting systems of equations are then solved numerically using spectral quasilinearization method (SQLM). The results reveal that the skin friction coefficient increases with increasing the values of nanoparticle volume fraction, unsteadiness and permeability parameters. The local Nusselt number reduces with increasing the value of nanoparticle volume fraction, Prandtl number and Eckert number. The local Sherwood number enhances with greater the value of nanoparticle volume fraction, unsteadiness, pressure gradient and chemical reaction parameters. Moreover, the method is checked against the previously published results and a very good agreement have been obtained.

Numerical Modelling on Flow Pattern Determination for Circular Staggered Cylinders in Crossflow

The flow of fluid nearby two circular staggered cylinders of identical size was investigated using computational flow dynamics (CFD). The center-to-center pitch ratio was taken constant, P/D = 2, and the incidence angle (°) was changing from 0° to 90°. Reynolds number (Re) used during the simulation was 300. This Re belongs to the periodic vortex shedding regime. Navier-Stokes equation was used to solve this unsteady flow problem in Two-dimensional incompressible form. Gambit 2.2.3 was used for the development of flow geometry configuration was as it allows imposing the boundary conditions of flow domain with ease. The flow field of staggered arrangement experimented evaluated experimentally in literature for five cases, namely, shear-layer-reattachment (SLR) for α = 0°, vortex pairing and enveloping (VPE) for α = 45°, vortex pairing, synchronized vortex shedding (SVS) and splitting for α= 90° was predicted using numerical model successfully.

Solution of Euler equations for unsteady flow problems using a central compact scheme

We solve the Euler equations in one and two dimensions using a central compact scheme coupled with an AUSM+ algorithm. The compact scheme is used in the form of a cell-face interpolation scheme to create the left- and right-states of the primitive variables required by the AUSM+ based solver. We invoke a TVD limiter to locally apply smaller stencil low-order dissipative formulae near shocks. No extra high-order artificial dissipation is added to stabilize the computation. A number of one and two dimensional test cases has been solved to show the effectiveness of this approach.

Extending the Meshless Local Petrov–Galerkin Method to Solve Stabilized Turbulent Fluid Flow Problems

The aim of this paper is to extend the meshless local Petrov–Galerkin method to solve stabilized turbulent fluid flow problems. For the unsteady incompressible turbulent fluid flow problems, the Spalart–Allmaras model is used to stabilize the governing equations, and the meshless local Petrov–Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. In this study, the moving least squares scheme interpolates the field variables. The proposed method solves three standard test cases of the turbulent flow over a flat plate, turbulent flow through a channel, and turbulent flow over a backward-facing step for evaluation of the method’s capability, accuracy, and validity purposes. Based on the comparison of the three test cases results with those of the experimental and conventional numerical works available in the literature, the proposed method shows to be accurate and quite implemental. The new extended method in this study together with the previously published works of the authors (on extending the meshless local Petrov–Galerkin method to solve laminar flow problems) now, for the first time, empower the meshless method to solve both laminar and turbulent flow problems.

Unsteady Stokes Flow through Porous Channel with Periodic Suction and Injection with Slip Conditions

This work is concerned with the influence of slip conditions on unsteady stokes flow between parallel porous plates with periodic suction and injection. The obtained unsteady governing equations are solved analytically by similarity method. The characteristics of complex axial velocity and complex radial velocity for different values of parameters are analyzed. Graphical results for slip parameter reveal that it has significant influence on the axial and radial velocity profiles. The effects of suction or injection are also observed. The problem of unsteady stokes flow through porous plates with no slip is recovered as a special case of our problem.

Unsteady Aerodynamic Modeling Based on POD-ARX

The lack of stability is a problem encountered when applying the classical POD-Galerkin method to problems of unsteady compressible flows around a moving structure. To solve this problem, a hybrid reduced-order model named POD-ARX is constructed in this paper. The construction of this model involves two steps, including first extracting the fluid modes with the POD technique and then identifying the modal coefficients with the ARX model. The POD modes with the block of all modified primitive variables are extracted from the system response to the training signal. Once the POD modes are obtained, the snapshots are projected on these modes to determine the time history of modal coefficients and the resulting modal coefficients are used to identify the parameters of ARX model. Then, the ARX model is used to predict the modal coefficients of the system response to the validation signal. Sample two-dimensional aerodynamic force calculations are conducted to demonstrate this method. Results show that this method can produce a stable and accurate prediction to the aerodynamic response with significant improvement of computational efficiency for linear and even some nonlinear aerodynamic problems. In addition, this method also shows good wide-band characteristics by using the “3211” multistep signal as the training signal.

Buoyancy Effects on MHD Unsteady Convection of a Radiating Chemically Reacting Fluid Past a Moving Porous Vertical Plate in a Binary Mixture

In this paper, the problem of unsteady magnetohydrodynamic free convective flow with thermal radiation and chemical reaction past a porous vertical plate moving through a binary mixture in an optically thin environment is investigated. The viscosity of the fluid is assumed to vary linearly with temperature. Due to the nature of corresponding dimensionless variable for temperature and its boundary condition as in the case of binary mixture, Boussinesq approximation and temperature dependent viscosity model were modified. The governing boundary layer equations are transformed using suitable similarity transformation and solved numerically. A parametric study of selected parameters is conducted and results for velocity, temperature, concentration, local skin friction coefficient, local Nusselt number and local Sherwood number are illustrated and physical aspects of the problem are discussed. Increasing the temperature dependent variable fluid viscosity leads to increase in the velocity of the fluid and heat transfer rate at the surface when Grashof related parameters are greater than one-tenth.

Time integration of unsteady nonhydrostatic equations with dual time stepping and multigrid methods

A time integration method based on dual time stepping coupled with a multigrid and implicit iterative solver is discussed for overcoming the temporal stiffness associated with modeling unsteady atmospheric flows. Adopting the dual time stepping allows solving an unsteady flow problem as a steady-state one by introducing a pseudo-time derivative into the governing equations. Consequently, the model uses a large time step and leads to a reduction in computational time. In the framework of the dual time stepping, a full approximation storage multigrid and lower–upper symmetric Gauss–Seidel implicit solver are employed to accelerate convergence in a structured, quadrilateral grid. The discretization of a physical-time derivative is based on a second-order backward differentiation formula or a family of singly diagonally implicit Runge–Kutta methods. The spatial terms of nonhydrostatic equations are discretized with a second-order Roe scheme in the finite volume method. To assess the performance of the model in terms of accuracy and efficiency, which are indicated by the normalized error norms and wall-clock time, respectively, three idealized tests are performed with the unsteady flow problems of isentropic vortex, rising thermal bubble and inertia-gravity waves. The results of numerical experiments consistently demonstrate that the dual time stepping coupled with explicit-first, singly diagonally implicit Runge–Kutta and multigrid methods is more efficient than the three-stage, third-order explicit Runge–Kutta method with comparable accuracy. The numerical experiment also indicates that the dual time stepping is more efficient when a stretched grid is employed. The efficiency in the framework of the dual time stepping is dependent on various numerical factors, including a multigrid level and cycle, Butcher coefficients, an implicit solver and a physical-time step.

Transforming projection methods to solve steady incompressible flows

Projection methods are known to be very effective in solving the unsteady flow problems. In this study the idea used in the SIMPLEC method is used to derive the general three- and the four-step projection methods. The concept used in the derivation of the projection methods is extended to obtain projection type methods for the solution of the steady flow problems directly, without resorting to a pseudo transient approach. The performance of the resulting three- and the four-step steady algorithms are compared with the well-known SIMPLE method in solving the steady flow problems on unstructured grids. The performance of the general three- and the four-step projection methods are also compared in solving the unsteady flow problems.

Prediction of Dynamic Stability Using Mapped Chebyshev Pseudospectral Method

A mapped Chebyshev pseudospectral method is extended to solve three-dimensional unsteady flow problems. As the classical Chebyshev spectral approach can lead to numerical instabilities due to ill conditioning of the spectral matrix, the Chebyshev points are evenly redistributed over the domain by an inverse sine mapping function. The mapped Chebyshev pseudospectral method can be used as an alternative time-spectral approach that uses a Chebyshev collocation operator to approximate the time derivative terms in the unsteady flow governing equations, and the method can make general applications to both nonperiodic and periodic problems. In this study, the mapped Chebyshev pseudospectral method is employed to solve three-dimensional periodic problem to verify the spectral accuracy and computational efficiency with those of the Fourier pseudospectral method and the time-accurate method. The results show a good agreement with both of the Fourier pseudospectral method and the time-accurate method. The flow solutions also demonstrate a good agreement with the experimental data. Similar to the Fourier pseudospectral method, the mapped Chebyshev pseudospectral method approximates the unsteady flow solutions with a precise accuracy at a considerably effective computational cost compared to the conventional time-accurate method.

Numerical solutions for unsteady boundary layer flow of a dusty fluid past a permeable stretching/shrinking surface with particulate viscous effect

PurposeThis present aims to present the numerical study of the unsteady stretching/shrinking flow of a fluid-particle suspension in the presence of the constant suction and dust particle slip on the surface.Design/methodology/approachThe governing partial differential equations for the two phases flows of the fluid and the dust particles are reduced to the pertinent ordinary differential equations using a similarity transformation. The numerical results are obtained using the bvp4c function in the Matlab software.FindingsThe results revealed that in the decelerating shrinking flow, the wall skin friction is higher in the dusty fluid when compared to the clean fluid. In addition, the effect of the fluid-particle interaction parameter to the fluid-phase can be seen more clearly in the shrinking flow. Other non-dimensional physical parameters such as the unsteadiness parameter, the mass suction parameter, the viscosity ratio parameter, the particle slip parameter and the particle loading parameter are also considered and presented in figures. Further, the second solution is discovered in this problem and the solution expanded with higher unsteadiness and suction values. Hence, the stability analysis is performed, and it is confirmed that the second solution is unstable.Practical implicationsIn practice, the flow conditions are commonly varying with time; thus, the study of the unsteady flow is very crucial and useful. The problem of unsteady flow of a dusty fluid has a wide range of possible applications such as in the centrifugal separation of particles, sedimentation and underground disposable of radioactive waste materials.Originality/valueEven though the problem of dusty fluid has been broadly investigated, limited discoveries can be found over an unsteady shrinking flow. Indeed, this paper managed to obtain the second (dual) solutions, and stability analysis is performed. Furthermore, the authors also considered the artificial particle-phase viscosity, which is an important term to study the particle-particle and particle-wall interactions. With the addition of this term, the effects of the particle slip and suction parameters can be investigated. Very few studies in the dusty fluid embedded this parameter in their problems.

Adjoint-based fluid dynamic design optimization in quasi-periodic unsteady flow problems using a harmonic balance method

Shape optimization in unsteady flow problems enables the consideration of dynamic effects on design. The ability to treat unsteady effects is attractive, as it can provide performance gains when compared to steady-state design methods for a variety of applications in which time-varying flows are of paramount importance. This is the case, for example, in turbomachinery or rotorcraft design. Given the high computational cost involved in time-accurate design problems, adjoint-based shape optimization is a promising option. However, efficient sensitivity analysis should also be accompanied by a significant decrease in computational cost for the primal flow solution, as well. Reduced-order models, like those based on the harmonic balance concept, in combination with the calculation of gradients via adjoint methods, are proposed for the efficient solution of a certain class of aerodynamics optimization problems. The harmonic balance method is applicable if the flow is characterized by discrete finite dominant flow frequencies that do not need to be integer multiples of a fundamental harmonic. A fully-turbulent harmonic balance discrete adjoint formulation based on a duality-preserving approach is proposed. The method is implemented by leveraging algorithmic differentiation and is applied to two test cases: the constrained shape optimization of both a pitching airfoil and a turbine cascade. A key advantage of the current approach is the accurate computation of gradients as compared to second order finite differences without any approximation in the linearization of the turbulent viscosity. The shape optimization results show significant improvements for the selected time-dependent objective functions, demonstrating that design problems involving almost-periodic unsteady flows can be tackled with manageable computational effort.

On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium

Unsteady two-dimensional flows of incompressible viscoelastic Maxwell medium with upper, low and corotational convective derivatives in the rheological constitutive law are considered. A class of partially invariant solutions is analyzed. Using transition to Lagrangian coordinates, an exact solution of the problem of unsteady flow near free-stagnation point was constructed. For the model with Johnson–Segalman convected derivative and special linear dependence of the vertical component of velocity, the general solutions were derived.

High-order detached eddy simulation of unsteady flow around NREL S826 airfoil

Delayed detached eddy simulation (DDES) has been a common approach for unsteady turbulent flow problems, but it has several weaknesses which limit its use to detached flows. On the other hand, some enhancements introduced recently have made it possible to apply to attached flows as well. In the present investigation, flow with relatively low Reynolds numbers of 100,000 and 145,000 around NREL S826 airfoil is simulated using DDES models with two different length scale definitions: standard length scale and shear-layer-adapted length scale (SLADDES). The solver, originally developed for aeroacoustic simulations, features fourth-order spatial accuracy enhanced with symmetry-preserving and dispersion-relation-preserving characteristics. Simulations are done at various angles of attack around stall regions. Results are compared with other simulations and measurements. The present results indicate that SLADDES accelerates transition from modeled turbulence mode to resolved turbulence mode even in attached flows. Consequently, SLADDES shows aerodynamic results in better agreement with the experiments than the standard DDES, and allows coarser grids thanks to low sensitivity to spanwise grid spacings. Moreover, it appears that both DDES and SLADDES approaches benefit from the low-dissipation, low-dispersion, and high-order scheme of the solver.

FGA-MMF method for the simulation of two-phase flows

PurposeThis paper aims to present a novel numerical technique for solving the incompressible multiphase mixture model.Design/methodology/approachThe multiphase mixture model contains a set of momentum and continuity equations for the mixture phase, a second phase continuity equation and the algebraic equation for the relative velocity. For solving continuity equation for the second phase and advection term of momentum, an improved approach fine grid advection-multiphase mixture flow (FGA-MMF) is developed. In the FGA-MMF method, the continuity equation for the second phase is solved with higher-order schemes in a two times finer grid. To solve the advection term of the momentum equation, the advection fluxes of the volume fraction in the continuity equation for the second phase are used.FindingsThis approach has been used in various tests to simulate unsteady flow problems. Comparison between numerical results and experimental data demonstrates a satisfactory performance. Numerical examples show that this approach increases the accuracy and stability of the solution and decreases non-monotonic results.Research limitations/implicationsThe solver for the multi-phase mixture model can only be adopted to solve the incompressible fluid flow.Originality/valueThe paper developed an innovative solution (FGA-MMF) to find multi-phase flow field value in the multi-phase mixture model. Advantages of the FGA-MMF technique are the ability to accurately determine the phases interpenetrating, decreasing the numerical diffusion of the interface and preventing instability and non-monotonicity in solution of large density variation problems.

Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems, two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries.

Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 2: numerical application

Abstract. The characteristic form of the Saint-Venant equations is solved in a stochastic setting by using a newly proposed Fokker–Planck Equation (FPE) methodology. This methodology computes the ensemble behavior and variability of the unsteady flow in open channels by directly solving for the flow variables' time–space evolutionary probability distribution. The new methodology is tested on a stochastic unsteady open-channel flow problem, with an uncertainty arising from the channel's roughness coefficient. The computed statistical descriptions of the flow variables are compared to the results obtained through Monte Carlo (MC) simulations in order to evaluate the performance of the FPE methodology. The comparisons show that the proposed methodology can adequately predict the results of the considered stochastic flow problem, including the ensemble averages, variances, and probability density functions in time and space. Unlike the large number of simulations performed by the MC approach, only one simulation is required by the FPE methodology. Moreover, the total computational time of the FPE methodology is smaller than that of the MC approach, which could prove to be a particularly crucial advantage in systems with a large number of uncertain parameters. As such, the results obtained in this study indicate that the proposed FPE methodology is a powerful and time-efficient approach for predicting the ensemble average and variance behavior, in both space and time, for an open-channel flow process under an uncertain roughness coefficient.

Bivariate spectral quasi-linearisation exploration of heat transfer in the boundary layer flow of micropolar fluid with strongly concentrated particles over a surface at absolute zero due to impulsive

The problem of unsteady micropolar fluid flow over a surface in which the heat energy falls at a lower limit of thermodynamic temperature scale due to impulsive is investigated. In this article, a new spectral method (BSQLM) for solving the partial differential equation is shown to unravel the heat transfer within the boundary layer. Some fluid layers at the free stream were given an impulsive motion in the horizontal direction. The thermal conductivity of the non-Newtonian fluid is assumed to be temperature dependent due to the influence of internal heat source; hence modified to suit the case of melting heat transfer following all the fundamental theories. The mathematical model was non-dimensionalised and parameterised using similarity transformation suitable to unravel the flow at short-time and long-time periods. Smooth transitions within the time frame 0 ≤ ξ ≤ 1 in the domain 0 ≤ η ≤ 1 are observed. The minimum temperature distribution is ascertained when the magnitude of Prandtl number is significantly large.