Duct Flow Problems Research Articles

Element-free Galerkin analysis of MHD duct flow problems at arbitrary and high Hartmann numbers

Element-free Galerkin analysis of MHD duct flow problems at arbitrary and high Hartmann numbers

Duct flow with leakage

The effect of wall leakage on the traditional duct flow equations based of the balance of mass, momentum and mechanical energy applied on a duct section is considered. Application of the momentum balance produces a Bernoulli equation type form, where the multiplier 1/2 in the dynamic pressure is however replaced by the value 1. Some relevant literature references are discussed. As an application, a nozzle duct flow problem is treated.

Uniformly convergent scheme for steady MHD duct flow problems with high Hartmann numbers on structured and unstructured grids

The steady incompressible Magnetohydrodynamic (MHD) duct flow problems with a transverse magnetic field at high Hartmann numbers (Ha) is a coupled system of convection-dominated diffusion equations and the solutions contain sharp boundary layers. We develop a five-point scheme and an edge-centered scheme on structured and unstructured grids with tailored finite point method (TFPM) for solving the coupled system of convection–diffusion equations. The five-point scheme is an upwind difference scheme, where the coefficients of the difference operator are tailored to some particular properties of the convection–diffusion equation with constant coefficients on the local cell. The edge-centered scheme is constructed by the continuity of the normal fluxes for each edge. The normal flux is expressed in terms of the edge-centered unknowns defined at the local cell and the unknowns at the adjacent cells are not needed. Uniform second order convergence can be obtained by the edge-centered scheme on structured and unstructured grids over a wide range of Ha 10 to 106. Both of the two schemes can correctly reproduce the fast change of the solutions which contain sharp boundary layers at high Ha even without refining the mesh. Numerical examples are provided to show the accuracy and of the TFPM.

The DRBEM solution of Cauchy MHD duct flow with a slipping and variably conducting wall using the well-posed iterations

The present study focuses on the numerical investigation of the Cauchy Magnetohyrodynamic (MHD) duct flow in the presence of an externally applied oblique magnetic field, with a slipping and variably conducting wall portion of the duct walls. The underspecified and overspecified boundary informations for the velocity of the fluid and the induced magnetic field on both slipping and variably conducting duct wall and its opposite part, respectively, constitutes the Cauchy MHD duct flow problem. This study aims to recompute the velocity of the fluid and induced magnetic field with specified slip length and conductivity constant, respectively, on the underspecified wall which is both slipping and variably conducting. The governing coupled convection-diffusion type MHD equations for the direct and inverse formulations are solved in one stroke using the dual reciprocity boundary element method (DRBEM). Both the velocity and induced magnetic field and their normal derivatives to be used as overspecified boundary conditions for the construction of Cauchy problem are obtained through the direct formulation of the problem. The well-posed iterations are used in the regularization of the ill-conditioned systems of linear algebraic equations resulting from the DRBEM discretization of Cauchy problem (inverse problem). Numerical solutions for the slip velocity and induced magnetic field are obtained for Hartmann number values $M$=5, 10, 50. The main advantages of the DRBEM are its boundary only nature and the capability of providing both the unknowns and their normal derivatives on the underspecified walls so that the conductivity constant and the slip length between them can be recovered at a low computational expense.

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.

Exponential high-order compact scheme on nonuniform grids for the steady MHD duct flow problems with high Hartmann numbers

In this article, we establish an exponential high-order compact (EHOC) difference scheme on non-uniform grids for the solution of the coupled equations representing the steady incompressible, viscous magnetohydrodynamic (MHD) flow through a straight channel of rectangular section. A main advantage of the non-uniform grids-based EHOC scheme is that it could use coarser mesh to capture the details within the computational domain for the MHD flow problems with high Hartmann numbers. Numerical experiments are carried out to validate the performance of the currently proposed scheme. Computation results of the MHD flow in the 2D square-channel problems are presented for Hartmann numbers ranging from 10 to 106. The numerical solutions obtained with the newly developed EHOC scheme are also compared with analytic solutions and numerical results by other available methods in the literature. All of these numerical results demonstrate that the currently proposed scheme is accurate, efficient and robust for the wide range of Hartmann numbers 10 to 106.

Wavelet optimized finite difference mesh for MHD flow in a circular duct

There is a large class of important problems in scientific and engineering applications where the solution shows very little variation in the core region but very rapid changes close to the boundary of the domain of interest. Magnetohydrodynamic flow problems belong to this category. Treatment of such problems numerically, therefore, requires a non-uniform mesh, which is very fine near the boundary. The determination of the size of the grid in different parts of the domain is, therefore, a challenging problem. Some recent studies suggest that wavelet methods can be effectively applied to create an optimized mesh. The method has mostly been used for one dimensional problems. The present paper investigates the problem for fully developed MHD duct flow problems which are basically two dimensional. The boundary layer character is more pronounced in one direction as compared to the other. The governing equations comprise a coupled system of two partial differential equations with Dirichlet boundary conditions. Wavelet based adaptive mesh is first created and then the system is discretized over this mesh. Computations have been carried out at different precision levels for various Hartmann numbers for a pipe of circular section. Contours are also drawn to depict the core and the boundary layer region.

An exponential compact difference scheme for solving 2D steady magnetohydrodynamic (MHD) duct flow problems

In this article, an exponential high-order compact (EHOC) difference scheme on the nine-point stencil is developed for the solution of the coupled equations representing the steady incompressible, viscous magnetohydrodynamic (MHD) flow through a straight channel of rectangular section. A key property of the EHOC scheme is that it has excellent stability and higher accuracy so that the high gradients near the boundary layer areas can be effectively resolved without refining the mesh. Numerical experiments are carried out to validate the performance of the currently proposed scheme. Computation results of the MHD flow in the 2D square-channel problems with different wall conductivities are presented for Hartmann numbers ranging from 10 to 106. The numerical solutions obtained with the newly developed EHOC scheme are also compared with analytic solutions and numerical results by other available methods in the literature.

A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers

AbstractIn this paper we propose a development of the finite difference method, called the tailored finite point method, for solving steady magnetohydrodynamic (MHD) duct flow problems with a high Hartmann number. When the Hartmann number is large, the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer. Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy. However, the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh. Firstly, we devise the tailored finite point method for the scalar inhomogeneous convection-diffusion problem, and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations. For each interior grid point of a given rectangular mesh, we construct a finite-point difference operator at that point with some nearby grid points, where the coefficients of the difference operator are tailored to some particular properties of the problem. Numerical examples are provided to show the high performance of the proposed method.

Initial Flow Rates for Ventilation Duct System Flow Analyses

The governing equations in duct system flow analyses are nonlinear, and iterative solution methods must be used. A suitable initial flow rate guess may be needed to start the calculations so that a converged solution can be finally achieved. A method to obtain a reasonably accurate initial flow rate guess is presented. In it the duct system is analyzed first in a strongly simplified setting assuming a linear (imaginary laminar) duct flow problem. The continuity equations at the junctions are satisfied. This first initial flow rate guess is then scaled to arrive at a more realistic final initial guess. The scaling is achieved by multiplication by a so–called balancing factor which is based on the application of mechanical energy balance for the whole duct system. The junction continuity equations are still satisfied for this final initial guess. A MATLAB program was generated to perform the calculations. Two demonstration example cases are presented to show the effect of the accuracy of the initial guess on the number of necessary iteration steps in connection with a PQ–duct system flow solution formulation.

Lattice Boltzmann modeling of pore‐scale fluid flow through idealized porous media

AbstractIn order to capture the hydro‐mechanical impacts on the solid skeleton imposed by the fluid flowing through porous media at the pore‐scale, the flow in the pore space has to be modeled at a resolution finer than the pores, and the no‐slip condition needs to be enforced at the grain–fluid interface. In this paper, the lattice Boltzmann method (LBM), a mesoscopic Navier–Stokes solver, is shown to be an appropriate pore‐scale fluid flow model. The accuracy and lattice sensitivity of LBM as a fluid dynamics solver is demonstrated in the Poiseuille channel flow problem (2‐D) and duct flow problem (3‐D). Well‐studied problems of fluid creeping through idealized 2‐D and 3‐D porous media (J. Fluid Mech.1959;5(2):317–328,J. Fluid Mech.1982;115:13–26,Int. J. Multiphase Flow1982;8(4):343–360,Phys. Fluids A1989;1(1):38–46,Int. J. Numer. Anal. Meth. Geomech.1999;23:881–904,Int. J. Numer. Anal. Meth. Geomech.2010; DOI: 10.1002/nag.898,Int. J. Multiphase Flow1982;8(3):193–206) are then simulated using LBM to measure the friction coefficient for various pore throats. The simulation results agree well with the data reported in the literature. The lattice sensitivity of the frictional coefficient is also investigated. Copyright © 2010 John Wiley & Sons, Ltd.

Two new upwind difference schemes for a coupled system of convection–diffusion equations arising from the steady MHD duct flow problems

In this paper, we develop two new upwind difference schemes for solving a coupled system of convection–diffusion equations arising from the steady incompressible MHD duct flow problem with a transverse magnetic field at high Hartmann numbers. Such an MHD duct flow is convection-dominated and its solution may exhibit localized phenomena such as boundary layers, namely, narrow boundary regions where the solution changes rapidly. Most conventional numerical schemes cannot efficiently solve the layer problems because they are lacking in either stability or accuracy. In contrast, the newly proposed upwind difference schemes can achieve a reasonable accuracy with a high stability, and they are capable of resolving high gradients near the layer regions without refining the grid. The accuracy of the first new upwind scheme is O( h + k) and the second one improves the accuracy to O( ε 2( h + k) + ε( h 2 + k 2) + ( h 3 + k 3)), where 0 < ε ≔ 1/ M ≪ 1 and M is the high Hartmann number. Numerical examples are provided to illustrate the performance of the newly proposed upwind difference schemes.

Adjoint wall functions: A new concept for use in aerodynamic shape optimization

The continuous adjoint method for the computation of sensitivity derivatives in aerodynamic optimization problems of steady incompressible flows, modeled through the k– ε turbulence model with wall functions, is presented. The proposed formulation leads to the adjoint equations along with their boundary conditions by introducing the adjoint to the friction velocity. Based on the latter, an adjoint law of the wall that bridges the gap between the solid wall and the first grid node off the wall is proposed and used during the solution of the system of adjoint (to both the mean flow and turbulence) equations. Any high Reynolds turbulence model, other than the k– ε one used in this paper, could also profit from the proposed adjoint wall function technique. In the examined duct flow problems, where the total pressure loss due to viscous effects is used as objective function, emphasis is laid on the accuracy of the computed sensitivity derivatives, rather than the optimization itself. The latter might rely on any descent method, once the objective function gradient has accurately been computed.

A bubble-stabilized least-squares finite element method for steady MHD duct flow problems at high Hartmann numbers

In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C 0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.

Continuous adjoint approach to the Spalart–Allmaras turbulence model for incompressible flows

A continuous adjoint formulation for the computation of the sensitivities of integral functions used in steady-flow, incompressible aerodynamics is presented. Unlike earlier continuous adjoint methods, this paper computes the adjoint to both the mean-flow and turbulence equations by overcoming the frequently made assumption that the variation in turbulent viscosity can be neglected. The development is based on the Spalart–Allmaras turbulence model, using the adjoint to the corresponding differential equation and boundary conditions. The proposed formulation is general and can be used with any other integral function. Here, the continuous adjoint method yielding the sensitivities of the total pressure loss functional for duct flows with respect to the normal displacements of the solid wall nodes is presented. Using three duct flow problems, it is demonstrated that the adjoint to the turbulence equations should be taken into account to compute the sensitivity derivatives of this functional with high accuracy. The so-computed derivatives almost coincide with “reference” sensitivities resulting from the computationally expensive direct differentiation. This is not, however, the case of the sensitivities computed without solving the turbulence adjoint equation, which deviates from the reference values. The role of all newly appearing terms in the adjoint equations, their boundary conditions and the gradient expression is investigated, significant and insignificant terms are identified and a study on the Reynolds number effect is included.

An enhanced thermal conduction model for the prediction of convection dominated solid–liquid phase change

An enhanced thermal conduction model for predicting convection dominated solid–liquid phase change is presented. The main feature of the model is to predict (1) the overall thermal behavior of the system and (2) the phase front position without recurring to the full solution of the Navier–Stokes equations. The model rests entirely on the conduction equation for both the solid and liquid phases. The effect of convection in the melt is mimicked via an enhanced thermal conductivity that depends on the dimensionless numbers and the geometry of the flow. The model is tested and confronted to full CFD solutions for a freezing duct flow problem and for buoyancy driven melting in an enclosure. In both cases, the predictions of the enhanced thermal conduction model show excellent agreement with that of the CFD model. Not only is the enhanced thermal conduction model simpler to implement but its simulations run at least ten times as fast as those of the CFD model. Consequently, the enhanced thermal conduction model is well suited for controlling real-time solid–liquid phase change processes that occur in industrial applications as well as in latent heat thermal energy storage systems.

Time‐domain BEM solution of convection–diffusion‐type MHD equations

AbstractThe two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Fundamental solution for coupled magnetohydrodynamic flow equations

In this paper, a fundamental solution for the coupled convection–diffusion type equations is derived. The boundary element method (BEM) application then, is established with this fundamental solution, for solving the coupled equations of steady magnetohydrodynamic (MHD) duct flow in the presence of an external oblique magnetic field. Thus, it is possible to solve MHD duct flow problems with the most general form of wall conductivities and for large values of Hartmann number. The results for velocity and induced magnetic field is visualized in terms of graphics for values of Hartmann number M ⩽ 300 .

A three-dimensional inverse forced convection problem in estimating surface heat flux by conjugate gradient method

A three-dimensional (3D) transient inverse forced convection problem is solved in the present study using the Conjugate Gradient Method (CGM) and the general purpose commercial code CFX4.2-based inverse algorithm to estimate the unknown boundary heat flux in a three-dimensional irregular duct flow problem. The advantage of calling CFX4.2 as a subroutine in the present inverse calculation lies in that many difficult but practical 3D inverse convection problem can be solved under this construction. Results obtained by using the conjugate gradient method to solve this 3D inverse forced convection problems are justified based on the numerical experiments. It is concluded that accurate boundary fluxes can be estimated by the conjugate gradient method except for the inlet surface and final time. The reason and improvement of this singularity are addressed. Finally, the effects of the height of duct, velocity of the inlet fluid and measurement errors on the inverse solutions are discussed.

Solution of the laminar duct flow problem by a boundary integral equation method

A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.