Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium

The boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium are considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the velocity parameter λ, which is the ratio of the external free stream velocity to the stretching surface velocity, permeability parameter of the porous medium k ₁, and Prandtl number Pr on the horizontal and transverse velocities, temperature profiles, surface heat flux and the wall shear stress, has been studied.     

p-version least squares finite element formulation for two-dimensional,incompressible fluid flow

A p-version least squares finite element formulation for non-linear problems is applied to the problem of steady, two-dimensional, incompressible fluid flow. The Navier-Stokes equations are cast as a set of first-order equations involving viscous stresses as auxiliary variables. Both the primary and auxiliary variables are interpolated using equal-order C⁰ continuity, p-version hierarchical approximation functions. The least squares functional (or error functional) is constructed using the system of coupled first-order non-linear partial differential equations without linearization, approximations or assumptions. The minimization of this least squares error functional results in finding a solution vector {δ} for which the partial derivative of the error functional (integrated sum of squares of the errors resulting from individual equations for the entire discretization) with respect to the nodal degrees of freedom {δ} becomes zero. This is accomplished by using Newton's method with a line search. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretching sheet

In the present work, the effect of MHD flow and heat transfer within a boundary layer flow on an upper-convected Maxwell (UCM) fluid over a stretching sheet is examined. The governing boundary layer equations of motion and heat transfer are non-dimensionalized using suitable similarity variables and the resulting transformed, ordinary differential equations are then solved numerically by shooting technique with fourth order Runge–Kutta method. For a UCM fluid, a thinning of the boundary layer and a drop in wall skin friction coefficient is predicted to occur for higher the elastic number. The objective of the present work is to investigate the effect of Maxwell parameter β, magnetic parameter Mn and Prandtl number Pr on the temperature field above the sheet.     

p-version least squares finite element formulation for two-dimensional,incompressible, non-Newtonian isothermal and non-isothermal fluid flow

This paper presents a p- version least squares finite element formulation (LSFEF) for two-dimensional, incompressible, non-Newtonian fluid flow under isothermal and non-isothermal conditions. The dimensionless forms of the diffential equations describing the fluid motion and heat transfer are cast into a set of first-order differential equations using non-Newtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature as well as the stresses and heat fluxes are interpolated using equal-order, C⁰-continuous, p-version hierarchical approximation functions. The application of least squares minimization to the set of coupled first-order non-linear partial differential equations results in finding a solution vector {δ} which makes the partial derivatives of the error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. The paper presents the implementation of a power-law model for the non-Newtonian Viscosity. For the non-isothermal case the fluid properties are considered to be a function of temperature. Three numerical examples (fully developed flow between parallel plates, symmetric sudden expansion and lid-driven cavity) are presented for isothermal power-law fluid flow. The Couette shear flow problem and the 4:1 symmetric sudden expansion are used to present numerical results for non-isothermal power-law fluid flow. The numerical examples demonstrate the convergence characteristics and accuracy of the formulation.     

A Fully—Implicit 2‐D Navier-Stokes Algorithm for Unstructured Grids

A fully-implicit algorithm is developed for the two-dimensional, compressible, Favre-averaged Navier-Stokes equations. It incorporates the standard k-? turbulence model of Launder and Spalding and the low Reynolds number correction of Chien. The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. A generalization of wall functions including pressure gradient effects is implemented to solve the near-wall region for turbulent flows using a separate algorithm and a hybrid grid. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, Trapezoidal or 3-Point Backward differencing. An incomplete LU factorization of the Jacobian matrix is implemented as a preconditioning method. The accuracy of the code and the efficiency of the solution strategy are presented for three test cases: a supersonic turbulent mixing layer, a supersonic laminar compression corner and a supersonic turbulent compression corner.     

Hydrodymagnetic three-dimensional flow over a stretching surface with heat and mass transfer

A general analysis has been developed to study the combined effect of the free convective heat and mass transfer on the steady three-dimensional laminar boundary layer flow over a stretching surface. The flow is subject to a transverse magnetic field normal to the plate. The governing three-dimensional partial differential equations for the present case are transformed into ordinary differential equation using three-dimensional similarity variables. The resulting equations, are solved numerically by applying a fifth order Runge-Kutta-Fehlberg scheme with the shooting technique. The effects of the Magnetic field Parameter M, buoyancy parameter N, Prandtl number Pr and Schmidt number Sc are examined on the velocity, temperature and concentration distributions. Numerical data for the skin-friction coefficients, Nusselt and Sherwood numbers have been tabulated for various parametric conditions. The results are compared with known from the literature.     

Fluid flow in curved ducts

The advent of standard algorithms for the numerical solution of partial differential equations has given researchers a new tool for fluid flow calculations. In this paper, single-phase flow in curved ducts is numerically simulated by imposing a spatially varying centrifugal force on a fluid flowing in a straight tube. The resulting set of partial differential equations is solved using the HARWELL-FLOW3D computer program. Comparison with other numerical and experimental results shows that this simplified formulation gives accurate results. The model neglects certain geometric terms of the order d/D, the duct-to-coil diameter ratio. The effect of these terms is investigated by considering the flow in a 90° bend for large d/D. It is shown that while there may be significant error in the prediction of the local variables for large d/D, the circumference-averaged quantities are well predicted.     

Finite volume computation of incompressible turbulent flows in general co-ordinates on staggered grids

A brief review of the computation of incompressible turbulent flow in complex geometries is given. A 2D finite volume method for the calculation of turbulent flow in general curvilinear co-ordinates is described. This method is based on a staggered grid arrangement and the contravariant flux componets are chosen as primitive variables. Turbulence is modelled either by the standard k–ε model or by a k–ε model based on RNG theory. Convection is approximated with central differences for the mean flow quantities and a TVD-type MUSCL scheme for the turbulence equations. The sensitivity of the method to the grid properties is investigated. An application of this method to a complex turbulent flow is presented. The results of computations are compared with experimental data and other numerical solutions and are found to be satisfactory.     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.     

Higher order spectral/hp finite element models of the Navier–Stokes equations

In this paper, we present higher order least-squares finite element formulations for viscous, incompressible, isothermal Navier–Stokes equations using spectral/hp basis functions. The second-order Navier–Stokes equations are recast as first-order system of equations using stresses as auxiliary variables. Both steady-state and transient problems are considered. For a better coupling of pressure and velocity, especially in transient flows, an iterative penalisation strategy is employed. The outflow-type boundary conditions are applied in a weak sense through the least-squares functional. The formulation is verified by solving various benchmark problems like the lid-driven cavity, backward-facing step and flow over cylinder problems using direct serial solver UMFPACK.     

Steady mixed convection boundary layer flow over a vertical flat plate in a porous medium filled with water at 4°C: case of variable wall temperature

The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated with water at 4°C (maximum density) when the temperature of the plate varies as x ^m and the velocity outside boundary layer varies as x ^{2 m} , where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel to a free stream velocity U _∞ oriented in the upward or downward direction, while the ambient temperature is T _∞ = T _m (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme. Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at 4°C (maximum density) compared to water at normal temperature.     

A stress‐based least‐squares finite‐element model for incompressible Navier–Stokes equations

In this paper we present a stress‐based least‐squares finite‐element formulation for the solution of the Navier–Stokes equations governing flows of viscous incompressible fluids. Stress components are introduced as independent variables to make the system first order. Continuity equation becomes an algebraic equation and is eliminated from the system with suitable modifications. The h and p convergence are verified using the exact solution of Kovasznay flow. Steady flow past a large circular cylinder in a channel is solved to test mass conservation. Transient flow over a backward‐facing step problem is solved on several meshes. Results are compared with that obtained using vorticity‐based first‐order formulation for both benchmark problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A least-squares method for solving the mixed form of the groundwater flow equations

The mixed form of the areal groundwater flow equations is solved with a least-squares finite element procedure (LESFEM). Hydraulic head and x- and y-directed fluxes are state variables. Physical parameters and state variables are approximated using a bilinear basis. Grid refinements and irregular domain boundaries are implemented on rectangular meshes. Residuals are constructed at collocation points for conservation of mass and Darcy's law. Boundary condition residuals are constructed at discrete points along the boundary. The residuals are weighted, squared and summed. A set of algebraic equations is formed by taking the derivatives of the weighted sum of the squares of the residuals with respect to each unknown parameter in the approximation for the state variable and setting them to zero. Proper choice of a potential scaling parameter and residual weights is essential for the effective application of the algorithm. Test problem results demonstrate that the method is effective for both transient and steady state cases. The LESFEM algorithm generates a C°-continuous velocity field. The continuous velocity field and the rectangular mesh simplify the implementation of algorithms that require tracking. In addition, rectangular meshes simplify mesh and boundary generation.     

A PRESSURE-BASED ALGORITHM FOR HIGH-SPEED TURBOMACHINERY FLOWS

The steady state Navier–Stokes equations are solved in transonic flows using an elliptic formulation. A segregated solution algorithm is established in which the pressure correction equation is utilized to enforce the divergence-free mass flux constraint. The momentum equations are solved in terms of the primitive variables, while the pressure correction field is used to update both the convecting mass flux components and the pressure itself. The velocity components are deduced from the corrected mass fluxes on the basis of an upwind-biased density, which is a mechanism capable of overcoming the ellipticity of the system of equations, in the transonic flow regime. An incomplete LU decomposition is used for the solution of the transport-type equations and a globally minimized residual method resolves the pressure correction equation. Turbulence is resolved through the k–ε model. Dealing with turbomachinery applications, results are presented in two-dimensional compressor and turbine cascades under design and off-design conditions. © 1997 John Wiley & Sons, Ltd.     

Performance of a multigrid calculation procedure in three-dimensional sudden expansion flows

The performance of a recently developed calculation procedure for steady incompressible flows is assessed in a variety of three-dimensional sudden expansion type flows representative of those encountered in several types of industrial equipment. The calculation procedure, called here BLIMM (for block-implicit multigrid method), is based on a coupled solution of the three-dimensional momentum and continuity equations in primitive variables, using the multigrid technique. Different Reynolds numbers and finite difference grids are considered for each flow situation. The rates of convergence and the computational times are reported for each case.     

A Mach‐uniform unstructured staggered grid method

A novel Mach‐uniform method to compute flows using unstructured staggered grids is discussed. The Mach‐uniform method is a generalization of the pressure‐correction approach for incompressible flows, and is valid for Mach numbers ranging from 0 (incompressible) to > 1 (supersonic). The primary variables (ρ u ,p and ρ) are updated sequentially. The grid consists of triangles. A staggered positioning of the variables is employed: the scalar variables are located at the centroids of the triangles, whereas the normal momentum components are positioned at the midpoints of the faces of the triangles. Discretization of the two‐dimensional flow equations on unstructured staggered grids is discussed. For the cell face fluxes there is a choice between first‐order upwind and central approximation. Flows around the NACA 0012 airfoil with freestream Mach numbers ranging from 0 to 1.2 are computed to demonstrate the Mach‐uniform accuracy and efficiency of the proposed method. Copyright © 2002 John Wiley & Sons, Ltd.     

Two‐dimensional prediction of time dependent,turbulent flow around a square cylinder confined in a channel

This paper presents two‐dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder–Sharma low‐Reynolds number k–ε model and a non‐linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×10⁴. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid‐point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third‐order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k–ε does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second‐order Crank–Nicolson scheme. The unsteady computations using the linear and non‐linear k–ε models significantly improve the velocity field predictions. However, the linear k–ε shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non‐linear k–ε model brings the two‐dimensional unsteady predictions of the time‐averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 John Wiley & Sons, Ltd.