Linear Source Term Research Articles

Numerical Analysis of Secondary Flow for Viscoelastic Fluid Through Elliptical Duct.

The flow behavior of viscoelastic non-Newtonian fluid in circular and non-circular ducts is of special engineering interest. Therefore a numerical analysis has been performed for viscoelastic non-Newtonian fluid in elliptical duct. Special attention is paid for the generation of secondary flow for laminar flow by using two kinds of constitutive equation, i.e., Maxwell and Reiner-Rivlin models. As for Maxwell model, body force caused by the elastic stress is approximated by linear source term. In calculation, viscosity was represented by adopting power-law fluid and boundary-fitted coordinate system was introduced as the method of coordinate transformation. The calculated results of two models show the secondary flow in elliptical duct as the same as theoretically analyzed by Green and Rivlin. Adding to the prediction of secondary flow, the generation mechanism of secondary flow has been argued by evaluating the production terms of the transport equation for streamwise vorticity. As a result of this examination, it was found that the term of viscous diffusion and the term containing second normal stress difference played an important role in producing the secondary flow near the wall. At the same time, it is interested phenomenon that the circular direction of secondary flow for viscoelastic fluid is opposite sigh to that of secondary flow for Newtonian turbulent flow. As its cause, the present study clarified that the term containing second normal stress difference of viscoelastic fluid is the same type equation for that of turbulence, while the sign of its term for viscoelastic fluid is opposite to that for turbulent flow.

A Field Monte Carlo Formulation for Calculating the Probability Density Function of a Single Scalar in a Turbulent Flow

The probability density function (PDF) formulation of one scalar field undergoing diffusion, turbulent convection and chemical reaction is restated in terms of stochastic fields. These fields are smooth in space as they have a length scale similar to that of the PDF. Their evolution is described by a set of stochastic partial differential equations, which are solved using a finite volume scheme with a stochastic source term. The application of this methodology to a particular flow is shown first for a linear source term, with exact analytical solution for the mean and standard deviation, and then for a nonlinear reaction.

Semi-implicit donor cell differencing and the hyperbolic conservation laws having source terms

Two semi-implicit versions of the first-order Courant, Isaacson and Rees upwind difference scheme for hyperbolic conservation laws having source terms are derived. Fourier analysis is used to analyse stability and phase accuracy of the schemes for a scaler conservation law having a linear source term. Numerical tests, in one and two dimensions, are performed for further insight and justification of their use in case the source term is non-linear. It is shown that the first variant enhances only stability while the second one improves both stability and phase accuracy. The second variant proves to be superior to the original scheme as well as our DOCTR and DOCSI (DOnor Cell differencing using TRapezoidal/SImpson's rule for source term integration) schemes (Ali and Kassar, Int. J. Modern Phys. C, Vol. 5, No. 2, 1994).

Convergence rate of approximate solutions to weakly coupled nonlinear systems

We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled 2 × 2 2 \times 2 systems appear, for example, in the context of resonant waves in gas dynamics equations. This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or L i p + \mathrm {Lip}^+ -stability) together with W − 1 , 1 W^{-1,1} -consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence rate estimates, both global (quantified in terms of the W s , p W^{s,p} -norms) and local. We focus our attention on the L i p + \mathrm {Lip}^+ -stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are L i p + \mathrm {Lip}^+ -stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of L i p + \mathrm {Lip}^+ -stability.

Application of Gradient Expansion to Non-Linear Gravitational Wave in Plane-Symmetric Universe

As an application of Gradient Expansion (long-wavelength) approximation, we studied the inhomogeneous universe including the gravitational wave(GW). For a plane-symmetric cosmological model, we could implement the 2nd order expansion and analyze the purely non-linear perturbation of GW. The result indicates that the non-linear effect plays an important role in anisotropic stage of the universe. We also confirm that the perturbation from the linear source term in gradient expansion is consistent with the result of linear perturbation.

Stability of solutions to a destabilized hopf equation

AbstractThe Hopf, or inviscid Burgers equation, destabilized by a linear source term is studied for periodic data. It is shown that for mean‐zero initial data whose integral has a unique minimum in each period, the stationary solution is a global attractor.

A Linearization for Secondary Flow Simulation of Viscoelastic Fluid.

For a translational symmetric case, the change rate of an elastic stress is approximated by the spin term alone. The contribution of the derived elastic stress to the momentum equation is decoupled from a velocity component in each coordinate. Then, the body force caused by the elastic stress is approximated by a linear source term within an iteration loop. The following cases are demonstrated : (1) a square duct, (2) a lid driven swirl flow in a cylinder, (3) a free surface movement of coaxial rotating cylinders.

On the structure of pressure fluctuations in simulated turbulent channel flow

Pressure fluctuations in a turbulent channel flow are investigated by analyzing a database obtained from a direct numerical simulation. Detailed statistics associated with the pressure fluctuations are presented. Characteristics associated with the rapid (linear) and slow (nonlinear) pressure are discussed. It is found that the slow pressure fluctuations are larger than the rapid pressure fluctuations throughout the channel except very near the wall, where they are about the same magnitude. This is contrary to the common belief that the nonlinear source terms are negligible compared to the linear source terms. Probability density distributions, power spectra, and two-point correlations are examined to reveal the characteristics of the pressure fluctuations. The global dependence of the pressure fluctuations and pressure-strain correlations are also examined by evaluating the integral associated with Green's function representations of them. In the wall region where the pressure-strain terms are large, most contributions to the pressure-strain terms are from the wall region (i.e., local), whereas away from the wall where the pressure-strain terms are small, contributions are global. Structures of instantaneous pressure and pressure gradients at the wall and the corresponding vorticity field are examined.

Rapid vaporization and heating of two parallel fuel droplet streams

An analysis is presented of the mechanisms of heat transfer and vaporization in an idealized two-dimensional fuel spray consisting of two parallel droplet streams injected in a hot gas. By assuming a constant relative velocity between the gas and liquid phases (inviscid approximation), a system of partial differential equations with linear operators and nonlinear source terms is obtained. With the use of Green's functions, this system is transformed into a group of non-linear ordinary differential equations and integral equations which are readily solved with numerical techniques. The dimensionless groups influencing the problem are identified and a discussion is presented of the effects of four relevant parameters including: a vaporization-diffusion time ratio, a liquid-gas mass flow ratio, a Spalding transfer number, and a Reynolds number. The method yields qualitatively correct estimates of the behavior of the gas and liquid phases which may serve as guidelines in the development of costlier numerical schemes. The method can be further exploited to include chemical reactions and to predict ignition of parallel droplet streams.

Oscillating Maxwellians: Exact solutions for the Boltzmann equation

Abstract For two classes of non linear Boltzmann equations, when the particles are of the Maxwell type and external forces are present, we present explicit distributions solutions. First, for the homogeneous one dimensional Kac model for which momentum conservation has been dropped, we include linear velocity forces and uniform source terms. Second, for the homogeneous and inhomogeneous Boltzmann equations of dimension d > 1, for which both energy and momentum conservations hold, we include either spatially dependent forces or velocity forces plus uniform sources. For both classes there exist examples of distributions which relax towards oscillating Maxwellians and other towards absolute Maxwellians. We give the complete proofs for the Kac model and present results for the d > 1 dimensional models.

Inhomogeneous similarity solutions of the Boltzmann equation with confining external forces

The Nikolskii transform makes it possible to construct inhomogeneous solutions of the Boltzmann equation from homogeneous ones. These solutions correspond to a gas in expansion, but if we introduce external forces, they can relax toward absolute Maxwellians. This property holds independently of the assumed intermolecular inverse power force. Consequently, for Maxwell molecules and from energy-dependent homogeneous distributions, we construct effectively a class of inhomogeneous similarity distributions with Maxwellian equilibrium relaxation. We review and investigate again the homogeneous distributions which can be written in closed form, for instance, we show that an elliptic exact solution proposed some years ago violates positivity. For Maxwell interaction with singular cross sections, we numerically construct inhomogeneous distributions having Maxwellian equilibrium states and study the Tjon overshoot effect. We show that both the sign and the time decrease of the external force as well as the microscopic model of the cross section contribute to the asymptotic behavior of the distribution. These inhomogeneous similarity solutions include a class of distributions that asymptotically oscillate between different Maxwellians. Two classes of external forces are considered: linear spatial-dependent forces or linear velocity-dependent forces plus source term.

Oscillating and absolute Maxwellians: Exact solutions for (d>1)-dimensional Boltzmann equations

A study of (d>1)-dimensional nonlinear Boltzmann equations is made for which both momentum and energy conservations hold. Maxwell particles in the presence of outside forces are assumed. For either linear spatial force or linear velocity force plus source term, a class of exact solutions for the homogeneous and inhomogeneous distributions is determined. In particular, a look is taken at oscillating external forces with a varying parameter and distributions relaxing towards oscillating Maxwellians are found. On the other hand, exact inhomogeneous distributions relaxing towards absolute Maxwellians are obtained. The exact solutions have asymptotic regimes that are solutions of the linear part of the Boltzmann equations and the relaxations towards these regimes are studied. Furthermore when equilibrium absolute Maxwellian states exist, whether the relaxation towards these states are from above or from below is investigated and the possibility of finding an overpopulation of high velocity particles at intermediate times is looked at.