NUMERICAL SOLUTIONS OF OPTIMAL CONTROL FOR THERMALLY CONVECTIVE FLOWS

We study the numerical solution of optimal control problems associated with two-dimensional viscous incompressible thermally convective flows. Although the techniques apply to more general settings, the presentation is confined to the objectives of minimizing the vorticity in the steady state case and tracking the velocity field in the non-stationary case with boundary temperature controls. In the steady state case we develop a systematic way to use the Lagrange multiplier rules to derive an optimality system of equations from which an optimal solution can be computed; finite element methods are used to find approximate solutions for the optimality system of equations. In the time-dependent case a piecewise-in-time optimal control approach is proposed and the fully discrete approximation algorithm for solving the piecewise optimal control problem is defined. Numerical results are presented for both the steady state and time-dependent optimal control problems. © 1997 John Wiley & Sons, Ltd.     

On the critical problem of F. D. pressure treatment for laminar flows confined by permeable walls

In the present work the viscous (low Reynolds) flow in plane ducts confined by permeable walls has been studied. A simple model of the filtrating walls has been used, with the normal velocity component proportional to the pressure jump across the wall, resulting in a non-standard boundary value Navier-Stokes problem. A critical analysis of the appropriate boundary condition and pressure problem has led to the conclusions of employing a simple explicit finite volume approach, and of avoiding the use of higher order finite difference schemes. In this paper a special emphasis on the structure of the involved computational matrices has been given to illustrate the chosen algorithm. The latter yields a steady state solution that is second order accurate in space, and it has an accuracy in time of order ≤ Δt (the time step), due to the explicit treatment of the velocity boundary conditions along the membrane. The model has been tested to study the effects of the inlet/outlet conditions, Reynolds number and filtrating wall constant.     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Computational intellectual analytical theory of computational analytical approach to rotating flow of non-Newtonian fluid

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Boundary element solution of unsteady magnetohydrodynamic duct flow with differential quadrature time integration scheme

A numerical scheme which is a combination of the dual reciprocity boundary element method (DRBEM) and the differential quadrature method (DQM), is proposed for the solution of unsteady magnetohydrodynamic (MHD) flow problem in a rectangular duct with insulating walls. The coupled MHD equations in velocity and induced magnetic field are transformed first into the decoupled time‐dependent convection–diffusion‐type equations. These equations are solved by using DRBEM which treats the time and the space derivatives as nonhomogeneity and then by using DQM for the resulting system of initial value problems. The resulting linear system of equations is overdetermined due to the imposition of both boundary and initial conditions. Employing the least square method to this system the solution is obtained directly at any time level without the need of step‐by‐step computation with respect to time. Computations have been carried out for moderate values of Hartmann number (M?50) at transient and the steady‐state levels. As M increases boundary layers are formed for both the velocity and the induced magnetic field and the velocity becomes uniform at the centre of the duct. Also, the higher the value of M is the smaller the value of time for reaching steady‐state solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Extremum problems of boundary control for steady equations of thermal convection

An inverse extremum problem of boundary control for steady equations of thermal convection is considered. The cost functional in this problem is chosen to be the root-mean-square deviation of flow velocity or vorticity from the velocity or vorticity field given in a certain part of the flow domain; the control parameter is the heat flux through a part of the boundary. A theorem on sufficient conditions on initial data providing the existence, uniqueness, and stability of the solution is given. A numerical algorithm of solving this problem, based on Newton’s method and on the finite element method of discretization of linear boundary-value problems, is proposed. Results of computational experiments on solving extremum problems, which confirm the efficiency of the method developed, are discussed.     

A promising boundary element formulation for three‐dimensional viscous flow

In this paper, a new set of boundary‐domain integral equations is derived from the continuity and momentum equations for three‐dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex‐variable technique is used to compute the divergence of velocity for internal points, while the traction‐recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd.     

Vibration analysis of a rod with complex boundary conditions

This paper deals with the forced longitudinal vibration of a rod carrying a concentrated mass and supported by a spring at one end. The vibration of the rod is excited by the motion of the support point at the other end. Since the boundary conditions of the problem are complex and it is necessary to consider the damping, we determine only the steady state periodic solution. First the linear system is analysed; then the material nonlinearity is considered and the approximate analytic solution of nonlinear partial differential equation with nonlinear boundary conditions is obtained by the perturbation method.     

Joint unidirectional motion of two viscous heat-conducting fluids in a tube

This paper studies an invariant solution of the problem of joint motion of two heat-conducting viscous immiscible fluids which have a common interface in a cylindrical tube under an unsteady pressure gradient. The problem reduces to a coupled initial-boundary-value problem for parabolic equations. A priori estimates of velocity and temperature perturbations are obtained. The steady state of the system is determined, and it is proved that if, in one of the fluids, the pressure gradient rapidly approaches zero, the perturbations of all quantities tend to zero. It is shown that if the pressure gradient has a nonzero limit, the solution reaches a steady state. In this case, the velocity field in the limit is the same as in conjugate Poiseuille flow, and the temperature is represented as a polynomial of the fourth order on the radial coordinate.     

Optimal flow control for Navier–Stokes equations: drag minimization

Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim, we handle with an optimal control approach applied to the steady incompressible Navier–Stokes equations. We use the Lagrangian functional approach and we consider the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover, we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier–Stokes equations. In this way, we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in the presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method. Copyright © 2007 John Wiley & Sons, Ltd.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Flow and heat transfer of an electrically conducting third grade fluid past an infinite plate with partial slip

The effects of partial slip on the steady flow and heat transfer of an electrically conducting, incompressible, third grade fluid past a horizontal plate subject to uniform suction and blowing is investigated. Two distinct heat transfer problems are studied. In the first case, the plate is assumed to be at a higher temperature than the fluid; and in the second case, the plate is assumed to be insulated. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. Numerical solutions for the governing nonlinear equations are obtained over the entire range of physical parameters. The effects of slip, magnetic parameter, non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. It is interesting to find that the velocity and the thermal boundary layers decrease with an increase in the slip, and as the slip increases to infinity, the flow behaves as though it were inviscid.     

Stick-slip flow of high density polyethylene in a transparent slit die investigated by laser Doppler velocimetry

The oscillating flow instability of a molten linear high-density polyethylene is carefully studied using a single screw extruder equipped with a transparent slit die. Experiments are performed using laser Doppler velocimetry in order to obtain the local velocities field across the entire die width. At low flow rate, the extrusion is stable and steady state velocity profiles are obtained. During the instability, the velocity oscillates between two steady state limits, suggesting a periodic stick-slip transition mechanism. At high flow rate, the flow is mainly characterized by a pronounced wall slip. We show that wall slip occurs all along the die land. An investigation of the slip flow conditions shows that wall slip is not homogeneous in a cross section of the slit die, and that pure plug flow occurs only for very high flow rates. A numerical computation of the profile assuming wall slip boundary conditions is done to obtain the true local wall slip velocity. It confirms that slip velocities are of the same order of magnitude as those measured with a capillary rheometer.     

Optimization of heat‐transfer rate into time‐periodic two‐dimensional Stokes flows

The periodic boundary displacement protocol leading to the optimum wall‐to‐fluid heat‐transfer rate, or to the most efficient mixing rate, in 2‐D annular Stokes flows is determined by calculating the steady periodic velocity and temperature fields. To obtain the steady periodic state one usually solves the dynamical system obtained after the spatial coordinates have been discretized. Here, we calculate the steady periodic state using an implicit method based on the discretization of the time coordinate over a period and the asymptotic regime is enforced by the periodicity condition in the computed temperature field. The obtained system of equations is solved using a Newton‐type iterative algorithm with invariant Jacobian. At each iteration step, the sparse linearized system is solved using a multi‐grid algebraic technique of rapid convergence. From a computational point of view and for the problem considered here, this method is an order of magnitude faster than the one based on a spatial discretization. Copyright © 2006 John Wiley & Sons, Ltd.     

On the Structure and Stability of the Set of Solutions of a Nonlocal Problem Modeling Ohmic Heating

In this work we analyze the existence, structure and stability of the set of positive solutions of a nonlocal elliptic boundary value problem modeling Ohmic heating. By regarding the total electric current flowing through the device by unit of cross-sectional area as the continuation parameter, we show that the set of steady states is constituted by a finite number of differentiable curves. Moreover, the instability index of a steady state changes by 1 when we pass through by a turning point and does not change if we pass through an hysteresis point. Some sufficient conditions so that the model admit a positive steady state for each value of the parameter, as well as for uniqueness and non-existence, will be also given.     

低Pr数流体非稳态自然对流流动的标度分析

施加了恒定热通量的竖直平板置于静止的均匀或线性分层流体中会在板壁上形成自然对流流 动. 它在达到稳态前将先历经起始和过渡两个阶段. 而表征其瞬时流动特征的主要参数是壁温、热边界层厚度、内层和外层速度边界层厚度、边界层内竖直方向的最大速度、以及层流边界层发展达到稳态所需的时间. 利用标度分析得到了均匀和线性分层低普郎特数(Pr<1)流体沿恒定热通量竖直平板的非稳态自然对流流动在各个发展阶段的标度关系.     

波流与结构物相互作用的数值模拟

采用高阶边界元法建立波流与任意形状结构物相互作用的时域数学模型。在小流速假设下,将速度势进行摄动展开,边界条件分解为双物体假设下的零阶波陡稳定问题和一阶波陡下的不稳定波浪问题。物面速度势及自由水面速度势的法向导数等未知量通过求解高阶边界元积分方程得到,而积分方程的求解则通过一个数值程序来实现。在每个时间步上采用四阶Runge-Kutta法更新下一时刻自由水面波面和速度势,自由水面上采用人工阻尼层消除散射波。通过波流与直立圆柱相互作用的数值计算研究了一阶激振力和二阶慢漂力随波数的变化关系以及圆柱周围波幅的分布曲线,并与已有频域结果和时域结果对比验证了所建模型的准确性,进而应用本文模型研究了波流与实际工程结构物相互作用的问题。     

Evolution of the joint motion of two viscous heat-conducting fluids in a plane layer under the action of an unsteady pressure gradient

A study is made of an invariant solution of the equations of a viscous heat-conducting fluid, which is treated as unidirectional motion of two such fluids in a plane layer with a common boundary under the action of an unsteady pressure gradient. A priori estimates of the velocity and temperature are obtained. The steady state is determined, and it is shown (under some conditions on the pressure gradient) that, at larger times, this state is the limiting one. For semiinfinite layers, a solution in closed form is obtained using the Laplace transform. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 94–107, July–August, 2008.     

Steady-State Analysis of Water Movement in a Semi-Infinite Unconfined Aquifer Under Constant Accretion

A partial analysis of the stream-aquifer interaction problem at steady state for a semi-infinite unconfined aquifer was presented by Bear (1979). This analysis was restricted to the case of influent boundary condition where the water flux is directed from the boundary into the aquifer. Here, we extend the analysis to provide analytical solutions for both influent and effluent conditions. Comparisons with numerical solutions are also presented for further validation.