Conservation Laws - MIT

1 Chapter 11 Conservation LawsIn the previous chapters we have studied numerical methods for ODEs. In the next couple of chapters we will developnumerical methods for partial differential equations (PDEs), which arise in many different physical Self-AssessmentBeforereading this chapter, you may wish to Differential Calculus [Reference to wherever this is taught] Divergence Theorem [Reference to wherever this is taught]Afterreading this chapter you should be able write Conservation laws in integral and differential form understand the behaviour of a convection equation understand the behaviour of a diffusion equation understand the behaviour of a convection-diffusion problem and how it varies with the Peclet numberRelevant self-assessment exercises:142 Conservation Laws in Integral and Differential FormIn most engineering applications, the physical system is governed by a set of Conservation laws.

2 For example, the gov-erning equations in gas dynamics correspond to the Conservation of mass, momentum, and energy. These conservationlaws are often written in integral form for a fixed physical domain. Suppose we have a physical domain, , with theboundary of the domain, . Then, the canonical Conservation equation assuming that the physical domain is fixed isof the form,ddtZ U dx+Z F(U) nds=Z S(U,t)dx,(75)whereUis the conserved state,Fis the flux of the conserved state,nis the outward pointing unit normal on theboundary of the domain, andSis a source Conservation law can be written as a partial differential equation by applying the divergence theorem whichstates that,Z F nds=Z Fdx.(76)Thus, Equation 75 becomes,5354ddtZ U dx+Z F nds=Z S dx,Z dUdtdx+Z F nds=Z S dx,Z U t+ F S dx= this last equation must be valid for any arbitrary domain, , this means that the integrand must be zero every-where, or, equivalently, U t+ F=S(77)Equation 77 is the Conservation law written as a partial differential of Mass for a Compressible FluidOne of the simplest examples of a Conservation law is the Conservation of mass for a compressible fluid.

3 Let thefluid density and velocity be (x,t)andv(x,t), respectively. The Conservation of mass for the fluid may be written inintegral form as:ddtZ dx+Z ( v) nds=0.(78)where (78) has the same form as Equation 75, with the conserved state,U= , flux,F= vand source,S=0. Thedifferential corresponding differential form for the Conservation of mass is: t+ ( v) =0(79)Example Equations for a Compressible FluidOften we wish to consider systems of Conservation laws. For example the Euler equations governing an inviscidcompressible flow correspond to the Conservation of mass, momentum, and energy of the fluid. The stateU, fluxF,and sourceSfor the two-dimensional Euler equations are,U= u v E F= u u2+p uv uH i+ v uv v2+p vH jS=0.(80)where =densityu=x-component of the velocityv=y-component of the velocityE=total energy per unit mass,p=static pressure,H=E+p =total enthalpy per unit conserved states are the density, ; x- and y- momenta, uand v; and total energy, E.

4 Thus, the first rowof 80 corresponds to the Conservation of mass, the second andthird rows correspond to the Conservation ofxandymomentum respectively, while the fourth row corresponds toconservation of energy. This system of equations is notquite complete, however, since the number of Conservation laws does not equal the number of dependent variables inthe equations. In particular, note that we have given four Conservation equations and the definition of total enthalpy,while the number of dependent variables is six ( ,u,v,p,E, andH). To complete the set of equations we define anequation of state. Often, we assume an ideal gas and use the ideal gas law. In terms of the dependent variables we have55introduced, the ideal gas law can be written as,p= ( 1) E 12 (u2+v2) ,(81)where is the ratio of specific heats (for air, ). You may be more familiar with the ideal gas law in the form,p= RTwhereRis the gas constant andTis the temperature.

5 Equation 81 is equivalent top= RTbut Equation 81is used sincep= RTintroduces a new dependent variable ( the temperature)and would therefore require yetanother state equation to complete the one-dimensional Burger s equation is given in differential form as: u t+u u x= is the conserved state and corresponding flux?(a)U=u,F=u(b)U=u,F=u2(c)U=12u,F=u2( d)U=u,F=12u243 ConvectionIn many applications, especially those in fluid dynamics, convection is the dominant physical transport mechanism overmuch of the domain of interest. While diffusion is always present, often its effects are small except in limited regions(often near solid boundaries where boundary layers form dueto the combined effects of convection and diffusion).In this section, we will derive the convection equation using the Conservation law as given in Equation 75. Specifi-cally, letUbe the conserved scalar quantity and let the fluxes be givenby,F=vU,S=0,(82)wherev(x,t)is the known velocity vector.

6 Note, a non-zero source term could be included, but for simplicity isassumed to be zero. This scalar Conservation law may be written as the first order partial differential equation U t+ (vU) =0.(83)Expanding the spatial derivatives gives, U t+v U+( v)U=0(84)Often a reasonable assumption is that the velocity field is divergence free ( v=0). In this case, we arrive at what iscommonly referred to as the convection equation, U t+v U=0.(85)Physically, this equation states that following along the streamwise direction ( convecting with the velocity), thequantityUdoes not developing numerical methods for convection-dominatedproblems, we will often rely on insight that can begained from the convection equation for the specific case when the velocity field is a constant value. Consider the flowwith constant velocity field, (x,t) =v. In this situation, the solution to Equation 85 has the following form,U(x,t) =U0( )where =x vt,(86)andU0(x)is the distribution ofUat timet=0.

7 By substitution, we can confirm that this indeed is a solution ofEquation 85. Using the chain rule, U t+v U= tU0( )+v xU0( )= U0 t+v U0 x where xand denote, respectively, the gradients with respect toxand . The partial derivatives of with respecttoxandtare, x =I, t= v,(87)whereIis the identity tensor. Upon substitution of these partial derivatives, U t+v U= U0 ( v)+v U0 I (88)= U0 ( v)+v U0 (89)=0.(90)Thus,U(x,t) =U0(x vt)is a solution to the convection ConvectionTo illustrate the behavior of the convection equation, we consider a simple one-dimensional convection problem U t+u U x=0(91)on the domain = [0,2], with constant flow velocityu=1. The initial distributionU0isU0(x) = (x )2 Figure 3 plots the initial solutionU0( att=0) as well asUatt= 15 Distribution ofUatt=0 and att=1 for a one-dimensional convection problem with velocityu= at the solution for the one-dimensional convectionproblem we observe thatU(t)has the same shape asUo, except the solution has been shifted by a distanceu t.

8 Using the definition of the total derivative,Uevolves intime as,dUdt= U t+ U this to Equation 91, we see that ifdxdt=u, thendUdt=0 ( ,U= constant). We call the linesx(t),such thatdxdt=u,characteristic linesor simplycharacteristics. Figure 43 shows the characteristic lines for the one-dimensional convection problem (in dashed lines) with the solutions att=0 andt=1 superimposed. As we can seefrom Figure 43, the solution at(x,t)is simply obtained by following the characteristic line back tot0and evaluatingthe initial 16 Characteristic lines for a one-dimensional convection problem with velocityu= may extend the idea of characteristics to any Conservation law which is a first order partial differential , we consider Conservation laws as in Equation 77 whereFandSmay be functions ofUbut not its Conservation laws exhibit convection like total derivative ofUis:dUdt= U t+ U dxdtAlong the linedxdt=dFdUthe total derivative ofUis:dUdt=S(U) F+ U dFdUdUdt=S(U)(92)which is an ordinary differential equation forU.

9 As in the simple one-dimensional convection problem, the linesx(t)which satisfydxdt=dFdUare known as the characteristic , that when there is a source term,Uis not constant along a characteristic line. However, we maymayevaluateUby simply integrating the ODE in Equation 92 along the characteristic line starting fromUo. The solutionat a particular point in space and timeU(x,t)depends only upon the points on characteristic line which goes through58the point(x,t). For any partial differential equation, we call the region which affects the solution at(x,t)thedomainof dependence. For convection, the domain of dependence for(x,t)is simply the characteristic line,x(t),s< the one-dimensional Burger s equation (given in exercise 1). What is the equation for thecharacteristics?(a)dxdt=u(b)dxdt=2u(c )dxdt=u2(d)dxdt=12u2 Exercise the one-dimensional Burger s equation on the domain[0,1]with initial conditionu0=x happens to the characteristic lines attincreases.

10 (a) The characteristics are parallel(b) The characteristics converge(c) The characteristics diverge(d) All of the above44 DiffusionIn many engineering applications the dominant physical transport phenomenon is modeled as diffusion. This sectionpresents the Conservation law for diffusion in differential form and discuss the behavior of problems modeled is characterized by a flux,F, of the form:F= Uwhere is the diffusion coefficient andUis the state. The Conservation law for the state,U, in a domain may bewritten as:ddtZ U dx+Z F nds=Z S(U,t)dx,The corresponding differential form is: U t ( U) =S(93)Equation 93 is a second-order partial differential equation often called the diffusion equation or heat equation. Partialdifferential equations of the form 93 arise in many applications including molecular diffusion and heat DiffusionWe now illustrate the behavior of the diffusion equation considering a simple one-dimensional model the one-dimensional diffusion equation with constant.