Unsteady Flow Problems Research Articles

A Unified Fractional-Step, Artificial Compressibility and Pressure-Projection Formulation for Solving the Incompressible Navier-Stokes Equations

AbstractThis paper introduces a unified concept and algorithm for the fractional-step (FS), artificial compressibility (AC) and pressure-projection (PP) methods for solving the incompressible Navier-Stokes equations. The proposed FSAC-PP approach falls into the group of pseudo-time splitting high-resolution methods incorporating the characteristics-based (CB) Godunov-type treatment of convective terms with PP methods. Due to the fact that the CB Godunov-type methods are applicable directly to the hyperbolic AC formulation and not to the elliptical FS-PP (split) methods, thus the straightforward coupling of CB Godunov-type schemes with PP methods is not possible. Therefore, the proposed FSAC-PP approach unifies the fully-explicit AC and semi-implicit FS-PP methods of Chorin including a PP step in the dual-time stepping procedure to a) overcome the numerical stiffness of the classical AC approach at (very) low and moderate Reynolds numbers, b) incorporate the accuracy and convergence properties of CB Godunov-type schemes with PP methods, and c) further improve the stability and efficiency of the AC method for steady and unsteady flow problems. The FSAC-PPmethod has also been coupled with a non-linear, full-multigrid and fullapproximation storage (FMG-FAS) technique to further increase the efficiency of the solution. For validating the proposed FSAC-PP method, computational examples are presented for benchmark problems. The overall results show that the unified FSAC-PP approach is an efficient algorithm for solving incompressible flow problems.

A unified coupling scheme between lattice Boltzmann method and finite volume method for unsteady fluid flow and heat transfer

In this paper, a unified coupling scheme between the lattice Boltzmann method (LBM) and the finite volume method (FVM) is proposed for the unsteady fluid flow and heat transfer problems. Three improvements are achieved comparing to the existing coupling methods. Firstly, a generalized form of the reconstruction operator (RO) is derived for the information transfer from macroscopic parameters to LBM distribution functions. The existing RO for various LBM can all be derived from this generalized form. Secondly, an RO corresponding to the incompressible LBM is derived to deal with the incompressible flow, which can prevent the inconsistency between the incompressible FVM and the density change in the LBM. Thirdly, the time coupling scheme between LBM and FVM is proposed for the unsteady simulations. The LBM and FVM are solving sequentially and the information is transferred between the two methods at the intervals. The coupling scheme is validated by three numerical examples: the convection–diffusion of a Gaussian pulse, the unsteady flow past a circular cylinder and the start-up process of the natural convection in a square cavity. The numerical results agree well with the existing researches. The flow past a porous medium is also simulated to show the application of the coupling method. This method can give the detailed flow information and save the computational time. The proposed coupling scheme improves and expands the coupling scheme between LBM and FVM. It has the flexibility to simulate unsteady multiscale processes.

Numerical analysis of unsteady MHD flow near a stagnation point of a two-dimensional porous body with heat and mass transfer, thermal radiation, and chemical reaction

The problem of unsteady MHD flow near a stagnation point of a two-dimensional porous body with heat and mass transfer in the presence of thermal radiation and chemical reaction has been numerically investigated. Using a similarity transformation, the governing time-dependent boundary layer equations for the momentum, heat and mass transfer were reduced to a set of ordinary differential equations. This set of ordinary equations were then solved using the spectral local linearization method together with the successive relaxation method. The study made among others the observation that the local Sherwood number increases with increasing values of the unsteadiness parameter and the Schmidt number. The fluid temperature was found to be significantly reduced by increasing values of the Prandtl number and the thermal radiation parameter. The velocity profiles were found to be reduced by increasing values of the chemical reaction and the Schmidt number as well as by the magnetic parameter.

Spectral Homotopy Analysis Method for PDEs That Model the Unsteady Von Kàrmàn Swirling Flow

A spectral homotopy analysis method (SHAM) is used to find numerical solutions for the unsteady viscous flow problem due to an infinite rotating disk. The problem is governed by a set of two fully coupled nonlinear partial differential equations. The method was originally introduced for solutions of nonlinear ordinary differential equations. In this study, its application is extended to a system of nonlinear partial differential equations (PDEs) that model the unsteady von Karman swirling flow. Numerical values of the pertinent flow properties were generated and validated against results obtained using the Keller-box numerical scheme. The results indicate that the present method is very accurate and can be used as an efficient tool for solving nonlinear PDEs of the type discussed in this paper.

The finite element method with heat transfer and fluid mechanics applications

1. The finite element method: introductory remarks 2. Some methods for solving continuum problems 3. Variational approach 4. Requirements for the interpolation functions 5. Heat transfer applications 6. One-dimensional steady-state problems 7. The two-dimensional heat conduction problem 8. Three-dimensional heat conduction applications 9. One-dimensional transient problems 10. Fluid mechanics finite-element applications 11. Use of nodeless degrees of freedom 12. Finite element analysis in curvilinear coordinates 13. Finite element modeling in annular passages 14. Extracting the F.E. domain from a larger flow system 15. Finite element application to unsteady flow problems 16. F.E.-based perturbation approach to unsteady flows.

Temperature field measurement of heated ventilation flow in a vehicle interior

The non–isothermal ventilation flow in a 1/10 scale vehicle interior model during compartment heating has been investigated experimentally. Three different ventilation modes were used to study their effect on the temperature field variation at the same Reynolds number. Instantaneous temperature field variations were measured by using a true colour image processing system and TLC (Thermochromic Liquid Crystal) particles. The relationship between the hue values of TLC colour image and the real temperatures was derived in advance and used to obtain real temperature variations in the vehicle interior. The ventilation flow in the passenger compartment was also visualised using a pulsed laser light sheet. As a result, the temperature field measurement technique used in this study was revealed to be a useful and powerful tool for analysing this kind of unsteady ventilation flow problem. The panel–vent heating mode is more effective for uniform temperature distribution in the rear compartment than the foot–vent mode, and the thermal non–uniformity could be minimised by using the hybrid–vent mode in a short elapse time.

ADAPTIVE SIMULATION OF THE INTERNAL FLOW IN A ROCKET NOZZLE

This work is a first step in the understanding of the interaction process between internal shock waves and the flow transition inside of a rocket nozzle that develops during the engine start-up phase or when the nozzle is operated at overexpanded conditions. In many cases, this transition in the flow pattern produces side loads in the nozzle due to an asymmetric pressure distribution on the wall, being harmful for the rocket’s integrity. To understand this phenomenon, a numerical simulation is performed by solving the three-dimensional Euler equations on unstructured tetrahedral meshes. With this model the computational cost to solve the equations significantly increases, therefore parallel processing is required. Also, an unsteady h-adaptive refinement strategy is used jointly with a Streamline Upwind Petrov-Galerkin and a discontinuity capturing scheme, both to keep the size of the fluid flow problem bounded and to sharply resolve the shock wave pattern. The mesh adaptation strategy is introduced. Since its performance is a major concern in the solution of unsteady flow problems, some implementation issues about the data structure chosen to represent the mesh are discussed. Average pressure distributions computed at the wall and the axis of the nozzle for various pressure ratios are analyzed based on experimental and numerical results from other authors.

Group solution for an unsteady non-Newtonian Hiemenz flow with variable fluid properties and suction/injection

The theoretic transformation group approach is applied to address the problem of unsteady boundary layer flow of a non-Newtonian fluid near a stagnation point with variable viscosity and thermal conductivity. The application of a two-parameter group method reduces the number of independent variables by two, and consequently the governing partial differential equations with the boundary conditions transformed into a system of ordinary differential equations with the appropriate corresponding conditions. Two systems of ordinary differential equations have been solved numerically using a fourth-order Runge—Kutta algorithm with a shooting technique. The effects of various parameters governing the problem are investigated.

A STUDY OF UNSTEADY FLOW THROUGH EARTH DAMS

Z_SOIL, commercial software based on finite element technique is proposed to simulate the unsteady seepage flow through earth dams, with variable upstream water heads. Experimental work, using Permeability and seepage model is conducted to verify the reliability and the accuracy of the numerical model. The accuracy and reliability of the model are adequate to handle unsteady flow problems with complex conditions. It is also demonstrated that the method of determining the seepage surface through the saturation described is natural, logical and practical in actual applications. The method provides a relatively simple computing scheme, as compared with traditional methods

Exact solutions for unsteady flow of second grade fluid generated by oscillating wall with transpiration

A problem of unsteady flow of a second grade fluid over flat plates with the impulsive and oscillating motion, starting from rest, and with the wall transpiration is considered. The exact solutions are derived by the Laplace transform, the perturbation techniques, and an extension of the variable separation technique together with similarity arguments. These solutions are written as the sum between the permanent solutions and the transient solutions. The variations of fluid behaviors with various physical parameters are shown graphically and analyzed. The results are validated by comparing the limiting cases of the present paper with the results of the related published articles.

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

SUMMARYA high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.

Flow and heat transfer characteristics of nanofluids in a rotating frame

The problem of unsteady MHD free convection flow of nanofluids via a porous medium bounded by a moving vertical semi-infinite permeable flat plate with constant heat source and convective boundary condition in a rotating frame of reference is studied theoretically. The velocity along the plate i.e. slip velocity is assumed to oscillate in time with constant frequency so that the solutions of the boundary layer are the same oscillatory type. The dimensionless governing equations for this investigation are solved analytically using small perturbation approximation. Two types of nanofluids, namely Cu–water and Al2O3–water are used. The effects of various parameters on the flow and heat transfer characteristics are discussed through graphs and tables.

Transient seepage analysis in zoned anisotropic soils based on the scaled boundary finite‐element method

SUMMARYThe scaled boundary finite‐element method, a semi‐analytical computational scheme primarily developed for dynamic stiffness of unbounded domains, is applied to the analysis of unsteady seepage flow problems. This method is based on the finite‐element technology and gains the advantages of the boundary element method as well. Only boundary of the domain is discretized, no fundamental solution is required and singularity problems can be modeled rigorously. Anisotropic and non‐homogeneous materials satisfying similarity are modeled with no additional efforts. In this study, firstly, formulation of the method for the transient seepage flow problems is derived followed by its solution procedures. The accuracy, simplicity and applicability of the method are demonstrated via four numerical examples of transient seepage flow – three of them are available in the literature. Homogenous, non‐homogenous, isotropic and anisotropic material properties are considered to show the versatility of the technique. Excellent agreement with the finite‐element method is observed. The method out‐performs the finite‐element method in modeling singularity points. Copyright © 2014 John Wiley & Sons, Ltd.

Stretching or shrinking sheet problem for unsteady separated stagnation-point flow

A detailed study of the problem of unsteady separated stagnation-point flow toward a stretching or shrinking sheet is presented. By use of a similarity transformation, the partial differential equations are reduced to a system of ordinary differential equations which are then solved numerically by use of the collocation method. The bvp4c Matlab program is used to perform the computation. This method is capable of finding multiple solutions for the flow reattachment case. The streamlines of the separation flow are presented for the cases when the origins of the stagnation flow and the stretching sheet are aligned and non-aligned. Comparison with results from the open literature reveals excellent agreement.

Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates

In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar–Gross–Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the implementation of the high-order compact finite-difference Lattice Boltzmann method (CFDLBM) in the generalized curvilinear coordinates is robust, efficient and accurate for solving steady and unsteady incompressible flows over practical geometries.

Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods—Part I: Formulation and application to the 2-D Burgers’ equation

The lattice Boltzmann method is being increasingly employed in the field of computational fluid dynamics due to its computational efficiency. Floating-point operations in the lattice Boltzmann method involve local data and therefore allow easy cache optimization and parallelization. Due to this, the cache-optimized lattice Boltzmann method has superior computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size and therefore the efficiency of the lattice Boltzmann method for steady flows. To quantify the computational performance of the lattice Boltzmann method for steady flows, a comparison study between the lattice Boltzmann method (LBM) and the alternating direction implicit (ADI) method was performed using the 2-D steady Burgers’ equation. The comparison study showed that the LBM performs comparatively poor on high-resolution meshes due to smaller time step sizes, while on coarser meshes where the time step size is similar for both methods, the cache-optimized LBM performance is superior. Because flow domains can be discretized with multiblock grids consisting of coarse and fine grid blocks, the cache-optimized LBM can be applied on the coarse grid block while the traditional implicit methods are applied on the fine grid blocks. This paper finds the coupled cache-optimized lattice Boltzmann-ADI method to be faster by a factor of 4.5 over the traditional methods while maintaining similar accuracy.

The HAM Solutions for Unsteady Boundary Layer Flow and Heat Transfer with Heat Source/Sink

Considering the combined effects of the magnetic field and viscous dissipation, this paper investigates the problem of two-dimensional incompressible unsteady flow over a horizontal continuous stretching sheet. Due to the strongly nonlinear and various parameters of this problem, the governing boundary layer equations are transformed into a system of nonlinear differential equations through the similarity transformation, and then analytical approximations of solutions are derived by homotopy analysis method. In addition, the effects of physical factors on the flow and heat transfer characteristics are examed and discussed graphically.

Unsteady Two-Dimensional Stagnation-Point Flow and Heat Transfer of a Viscous, Compressible Fluid on an Accelerated Flat Plate

The general formulation and exact solution of the Navier–Stokes and energy equations regarding the problem of steady and unsteady two-dimensional stagnation-point flow and heat transfer is investigated in the vicinity of a flat plate. The plate is moving at time-dependent or constant velocity towards the main low Mach number free stream or away from it. The main stream impinges along z-direction on the flat plate with strain rate a and produces two-dimensional flow. The fluid is assumed to be viscous and compressible. The density of the fluid is affected by the existing temperature difference between the plate and potential far field flow. Suitably introduced similarity transformations are used to reduce the governing equations to a coupled system of ordinary differential equations. Finite Difference Scheme is used to solve these non-linear ordinary differential equations. The obtained results are presented over a wide range of parameters characterizing the problem. It is revealed that the significance of the increase of thermal expansion coefficient, β, and wall temperature on velocity and temperature distributions is much more noticeable for a plate moving away from impinging flow. Moreover, negligible shear stress and heat transfer is reported between the plate and fluid viscous layer close to the plate for a wide range of β coefficient when the plate moves away from incoming far field flow.

Thermal effects in Stokes’ Second Problem for Unsteady Second Grade Fluid Flow through Porous Medium

In this paper, we studied the effect of thermal in Stokes' second problem for unsteady second grade fluid flow through porous medium. The expressions for the velocity field and the temperature field are obtained analytically. The effects of various emerging parameters on the velocity field and temperature field are studied through graphs in detail.

On an Interpolation Based Spectral Homotopy Analysis Method for PDE Based Unsteady Boundary Layer Flows

This work presents a new approach to the application of the spectral homotopy analysis method (SHAM) in solving non-linear partial differential equations (PDEs). The proposed approach is based on an innovative idea of seeking solutions that obey a rule of solution expression that is defined in terms of bivariate Lagrange interpolation polynomials. The applicability and effectiveness of the expanded SHAM approach are tested on a non-linear PDE that models the problem of unsteady boundary layer flow caused by an impulsively stretching plate. Numerical simulations are conducted to generate results for the important flow properties such as the local skin friction. The accuracy of the present results is validated against existing results from the literature and against results generated using the Keller-box method. The preliminary results from the proposed study indicate that the present method is more accurate and computationally efficient than more traditional methods used for solving PDEs that describe nonsimilar boundary layer flow.