The numerical computation of turbulent flows

The paper reviews the problem of making numerical predictions of turbulent flow. It advocates that computational economy, range of applicability and physical realism are best served at present by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy $\text{k}$ and its dissipation rate ?, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour. The width of applicability of the model is demonstrated by reference to numerical computations of nine substantially different kinds of turbulent flow.     

An algebraic stress model for stratified turbulent shear flows

Utilizing a simple time dependent one dimensional example as a test case this paper discusses a solution which represents the important characteristics of a bouyancy dominated shear flow by solving four partial differential equations in addition to the mean equations of motion. This suggested model solves equations for total turbulent kinetic energy, $\text{k}$, total turbulent temperature fluctuations, $\text{k}{}_{\mathrm{t}}$, eddy dissipation, ?, and thermal eddy dissipation, ?_$\text{t}$. Three separate versions of this model are discussed—an algebraic length scale version, a Prandtl-Kolmogorov eddy viscosity version, and an algebraic stress and heat flux model. The final version (requiring six partial differential equations) manages to replicate results for a much more complicated version (requiring ten partial differential equations). The advantages for two and three dimensional problems are even greater.     

Viscous flow over a cone at moderate incidence—I: Hypersonic tip region

A numerical investigation of the three-dimentsional viscous flow over a slender cone at moderate incidence is present; $\text{α/θ ≤ 2}$, where α is the angle of attack and θ is the half-cone vertex angle. In Part I presentd herein, the flow properties within the strong interaction and tip merged regions are discussed. The theoretical analysis relies on the “single layer” model of Rudman and Rubin[6] and the numerical analysis is based on a predictor-corrector scheme that has been recently developed by the present authors[8, 10]. The influence of the streamwise pressure gradient, $\text{P}{}_{\mathrm{x}}$, on the governing equations, and the implications, on upstream influence and free interaction, of various-difference quotients for representing $\text{P}{}_{\mathrm{x}}$ are discussed. Details of the flow properties including shock location, heat transfer, static pressure, surface flow patterns, the entropy distribution, as well as the approach to conical flow conditions are presented. The effects of diffusion on the inviscid vortical singularity are examined and a “lift-off” phenomena is observed for $\text{α/θ ≥ 1}$.     

A numerical method for predicting natural convection in horizontal cylinders with asymmetric boundary conditions

A numerical technique is developed to predict the two-dimensional transient natural convection heat transfer within a horizontal cylinder. Finite difference analogs of the Navier-Stokes and thermal energy equations are solved in the stream function-vorticity framework. The solution method, which is a modification of an alternating-direction implicit (ADI) scheme wherein the convective terms are evaluated explicitly, is found to be computationally more efficient than either an ADI or an explicit method. Unlike previous work, the present technique will accommodate completely arbitrary temperature boundary conditions. Thus, rather than considering an annular space or half of a symmetric cylinder, the solutions are determined for a full cylinder. A Cartesian form of the governing equations is employed at the point $\text{r= 0}$ where the polar coordinate equations become singular. The computed results are found to be in good agreement with previously published experimental data.     

New finite element results for the square cavity

New results for the recirculating flow inside a square cavity obtained by a finite element method are presented. The full Navier-Stokes equations in the form of a single, fourth order equation for stremfunction is recast into arestricted variational principles, which form finite element discretization. A triangular element with Hermitian interpolation is used, such that the velocities are continuous and the incompressibility is satisfied exactly. The resulting nonlinear system is solved by Newton-Raphson iteration. Calculations are carried out with several gridworks of progressive show: (1) the convergence of solutions with refinement for fixed Reynolds number $\text{R}$; and (2) the loss of accuracy with $\text{R}$ for fixed gridwork. The range of $\text{R}$ covered is from $\text{10}{}^{\mathrm{?4}}$ to 3450. The features illustrated include the enlargement of the inviscid core of rigid rotation, the intensification of primary and secondary vortices and the appearance of a third secondary vortex near the upper upstream corner at $\text{R =1500}$.     

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in $H^1$-norm and for pressure in $L^2$-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.     

Iterative numerical solutions and boundary conditions for the parabolized Navier-Stokes equations

A new numerical iteration scheme for solving the parabolized Navier-Stokes (PNS) equations is presented. This scheme has all the features and advantages of the successive line over relaxation (SLOR) technique, and thus it can be easily accelerated to get much higher rate of convergence of the global iteration scheme than previously suggested schemes. The choice of appropriate downstream boundary conditions for the PNS and Navier-Stokes equations is discussed in the context of boundary layer simulation. A critical comparison of accuracy and rate of convergence is performed for the flow over a flat plate.     

Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.     

On the use of symmetrizing variables for vacuums

The paper is devoted to the computation of shallow-water equations (or Euler equations) in using an approximate Godunov scheme called VFRoe, when the flow may include dry areas (or very low density regions). This is achieved with the help of symmetrizing variables. Overall we are able to insure the discrete preservation of positive variables on interfaces, and at the same time to compute vacuum occurence or propagation of shock waves over a near-vacuum. A short section is also dedicated to the non-conservative hyperbolic equations arising within the setting of one-equation or two-equation turbulent compressible models. Many numerical tests confirm the capabilities of the scheme, and measuring the L¹ error norm in particular cases enables us to specify the actual rate of convergence.     

Numerical integration of a coupled Korteweg-de Vries system

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for an arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence, which gives the conditions and the appropriate choice of the grid sizes. The method is applied to the Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to nonintegrable cases.     

Nonlinear superposition principles: A new numerical method for solving matrix Riccati equations

Previously derived superposition formulae, expressing the general solution of the matrix Riccati equation $\text{W}\text{?}\text{W(t) = A + WB + CW + WDW (A(t), B(t), C(t), D(t), W(t) ?}\text{R}{}^{\mathrm{n\times n}}\text{, n ≥ 2, t ?}\text{R}\text{)}$ in terms of 5 particular solutions, are applied to obtain numerical solutions of this coupled system of nonlinear equations.     

Sensitivity of the numerical analysis of the three-fluid plasma mixed initial-boundary value problem

The sensitivity of the numerical solution of the nonlinear three-fluid equations governing the effect of forcing a time dependent disturbance at a point in a plasma is investigated. With the equations transformed into a diagonal form it is shown that only certain variables may be prescribed as functions of time at $\text{x=0}$, where these specified functions must satisfy certain compatibity conditions. With forward differences used to replace the time derivative and either forward or backward differences used to replace the spatial derivatives, the difference equations formulated are consistent and their solution converges to the solution of the differential equations. This convergence is true as long as the domain of dependence concept is adhered to. A lineary analysis provides a guide to the actual stability of the system of equations. From this analysis it is seen that the magnitude of the collission frequences, as well as the speed of light, restricts the size of the steps which may be used. Furthermore, it is shown that the solution is extremely sensitive to the boundary and initial conditions specified.     

Comparison of adaptive controllers

The paper investigates adaptive control of discrete-time processes satisfying first-order linear difference equations with random coefficients which may be constant or time-varying. Structural relationships between sub-optimal adaptive control laws are discussed and results of systematic Monte-Carlo simulations are reported. These lead to a comparison of two sub-optimal adaptive controls ($\text{‘}\text{optimal}\text{-k-}\text{step-ahead}$’ and ‘self-tuning’) and to conclusions about the need for and effectiveness of adaptive control of the systems simulated. It is conjectured that the results and the classification scheme they suggest might have more general validity.     

System identification in nonlinear structural seismic dynamics

This paper deals with the development of efficient identification methods for the determination of parameters associated with the nonlinear response of structural systems subject to seismic conditions. The equation of motion of the system is given by $\text{M}\text{u}\text{?}\text{+ h(/.u, u, a) = p(t)}$, where the external force vector $\text{p}$ and the mass matrix $\text{M}$ are assumed to be known. Also given is an observation vector $\text{w}$, consisting of some or all of the components of $\text{u}$, the vector of the nodal displacements. The vector function $\text{h}$, denoting the restoring forces of the mechanical system, is parametrically given in terms of a constant vector $\text{a}$. Three different methods are presented for the determination of the parameters governing the vector function $\text{h}$. The first one is a direct approach requiring the unknown coefficients to appear linearly in the model equation. The remaining two are based on methods of control and optimization theory, and are exempt from the limitations of the direct method. The relative merits of each approach are discussed and extensive numerical experimentation is presented. Only numerical results for one degree of freedom systems are reported in this paper.     

Optimal input design for an AR model with output constraints

The output power constraint problem of optimal input design for parameter estimation for an autoregressive model is considered. A simple method to obtain an optimal design by solving two sets of $\text{p}\text{2-}\text{linear}$ simultaneous equations and a polynomial equation of $\text{p}\text{2}\text{th}$ order is proposed and two nontrivial examples are given to illustrate this methodology.     

Multi-search optimization techniques

Two multi-search optimization techniques are developed. The motive is to show that increasing the search directions results in a better rate of convergence. This idea was first investigated by Miele and Cantrell [1]. Their technique is computationally attractive, but exact line-search is needed in two directions. Therefore the problem of line-search has been more complicated since at the end of each iteration two stopping functions have to satisfied. Also, built-in safeguards were used to ensure the stability of the descent process. For nonquadratic functions Miele's method (as well as many other optimization techniques) needs restarting every $\text{n}$ + 1 iterations in order to improve the convergence rate. Extending this two-direction search of Miele to an $\text{n-}\text{direction}$ search in order to further improve the rate of convergence would highly complicate the search process, since exactly $\text{n}$ line-searches are needed together with $\text{n}$ simultaneous stopping functions to be satisfied besides the resulting increase in the built-in safeguards to impractical limits. In the techniques presented the optimum step-size in each direction is obtained by simply minimizing a quadratic function of a linearized gradient, and only sufficient decrease is needed along the search directions without the need of exact line-searches. Instead, simple successive halving is used till the function starts to increase; then a new iteration is started without the use of any simultaneous stopping functions at the end of each iteration. Also, the $\text{n-}\text{direction}$ case has been developed in exactly the same fashion as the two-direction case without quiring the use of $\text{n}$ simultaneous stopping functions a property which is not possible if the method of Miele is to be extended to the $\text{n-}\text{direction}$ case. The highly effective rate of convergence obtained by using the $\text{n-}\text{direction}$ search really justifies the would-be increase in the required computational effort. Furthermore, no restarting has been needed to improve the convergence rate. The results of two tried examples show that the multi-search optimization techniques that have been developed have excellent convergence rates without endangering the stability requirement.     

Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term

In this article, we study and analyze a Galerkin mixed finite element (MFE) method combined with time second-order discrete scheme for solving nonlinear time fractional diffusion equation with fourth-order derivative term. We firstly introduce an auxiliary variable $\sigma =△u$, reduce the fourth-order problem into a coupled system with two equations, discretize the obtained coupled system at time $t_{k?\frac{\alpha }{2}}$ by a second-order difference scheme with second-order approximation for fractional derivative, then formulate mixed weak formulation and fully discrete MFE scheme. Further, we give the detailed proof for stability of scheme, the existence and uniqueness of MFE solution, and a priori error estimates. Finally, by some numerical computations, we test the theoretical results, which illustrate that we can obtain the numerical results for two variables, moreover, we arrive at second-order time convergence orders, which are higher than the ones yielded by the $L1$-approximation.     

Optimally Convergent HDG Method for Third-Order Korteweg–de Vries Type Equations

We develop and analyze a new hybridizable discontinuous Galerkin method for solving third-order Korteweg–de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton–Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.     

Efficient decomposition of separable algebras

We present new, efficient algorithms for computations on separable matrix algebras over infinite fields. We provide a probabilistic method of the Monte Carlo type to find a generator for the center of a given algebra $\text{A}\text{⊆}\text{F}{}^{\mathrm{m\times m}}$ over an infinite field $\text{F}$. The number of operations used is within a logarithmic factor of the cost of solving m×m systems of linear equations. A Las Vegas algorithm is also provided under the assumption that a basis and set of generators for the given algebra are available. These new techniques yield a partial factorization of the minimal polynomial of the generator that is computed, which may reduce the cost of computing simple components of the algebra in some cases.