Advection, Conservation, Conserved Physical Quantities, Wave Equations

Transcription

1 EP711 Supplementary Material Thursday, September 4, 2014 Advection, Conservation, Conserved Physical Quantities, Wave Equations Jonathan B. Snively!Embry-Riddle Aeronautical University

2 Contents EP711 Supplementary Material Thursday, September 4, 2014 Lagrangian vs. Eulerian Coordinates Advection and Conservation Conserved Physical Quantities Wave Equations

3 Lagrangian Description of Fluid Motion Describes temporal evolution of a quantity Q in a parcel of gas or fluid. Assume Q(x(t),t), where x(t) may be a vector position and t is time, tracking the evolution of a single parcel s properties. Effectively, Q is a function of time only. Conservation of Q within the parcel is expressed by: dq dt =0 Which, simply, states that some quantity Q in a parcel does not vary in time the total derivative of Q is zero.

4 Total Derivative Since Q is a function of x(t) and t, the Lagrangian total derivative, may be expanded with help from the chain rule, yielding the following expression: dq @t rq Considering that x/ t=v (velocity), dq dt + ~v Thus, the material derivative can be expanded as: d + ~v r

5 Total Derivative In case you are having doubts about the total derivative and the chain rule, let s write out the full expression, where x=(x(t), y(t), z(t)): dq(x, y, z, t) @x Or, Where dq @t rq ˆk

6 Eulerian Description of Fluid Motion Describes temporal evolution of quantity Q in a gas or fluid at fixed points in space. Conservation of Q in space and time + ~v rq =0 This states that a quantity Q varies in space and time, while it is advected (transported) by a fluid velocity field v. This is also known as an advection equation, which here is not expressed in its most-general form.

7 Contents EP711 Supplementary Material Thursday, September 4, 2014 Lagrangian vs. Eulerian Coordinates Advection and Conservation Conserved Physical Quantities Wave Equations

8 Advection Equation for Non-Divergent Flow A more general conservative advection equation, describing conservation of a quantity in a divergent flow, is given + r (Q~v) =0 r (Q~v) =~v rq + Qr ~v In contrast, the previous expressions, given by: dq dt + ~v rq have assumed non-divergent velocity: r ~v =0

9 Conservation Laws and Continuity Equations This more general advection equation can be considered a continuity equation or conservation law for a quantity Q that is conserved under all conditions for example, mass density, momentum, + r (Q~v) =0 Here, the quantity in parentheses is known as the flux f, such that the expression can be = r ~f Although Q is here a scalar, it does not need to be!

10 Conservation Laws and Continuity Equations fx Q dz fx + ( fx/ x) dx = r ~f

11 Flux Form (for Conservative Advection) For a scalar quantity (such as density), the flux f depends on Q(x,t), such that the expression can be = r ~f Where Q(x,t) is a vector, this expression effectively contains one equation for each spatial dimension; f is then a tensor. An example we will discuss is the momentum conservation equation, here shown using somewhat ( ~v) = r ( v iv j + p ij )

12 Just a note... The advection equation is a form of a one-way wave equation, with solutions traveling at velocity v. Combine two of them to obtain the familiar second-order wave equation!

13 Contents EP711 Supplementary Material Thursday, September 4, 2014 Lagrangian vs. Eulerian Coordinates Advection and Conservation Conserved Physical Quantities Wave Equations

14 Conservation of Mass For a scalar quantity (such as density), the flux f depends on Q(x,t), such that the expression can be = r ~f The best-known conservation law is the continuity equation for neutral = r ( ~v) @x ( v)

15 Conservation of Charge (Same!) A vector current density is indicative of a flux of scalar charge density at a certain drift velocity: The resulting conservation law is the continuity equation for electric = r ( ~v) Integral form:

16 Conservation of Momentum Momentum equations for each spatial direction (x, y, z for Cartesian coordinates) here contained in one expression as a vector ( ~v) = r ( v iv j + p ij ) This expression effectively describes Newton s law. In one dimension, momentum conservation can ( ( v2 + p)

17 Conservation of Energy Energy is a scalar quantity = r {(E + p)~v} Here, it is important to define how E relates to other quantities via an equation of state We will assume an ideal single-constituent gas. E = (~v ~v) Internal + Kinetic The internal energy is given by: = c v T = p ( 1)

18 Euler Equations of Motion: Example equations for a simple atmosphere supporting acoustic and gravity waves, but (lazily) excluding viscosity and thermal conduction (e.g., Snively s 2003 MS thesis): We model the atmosphere as a non-rotating, fully-nonlinear, compressible gas: Mass Density: Momentum: Energy: + ( v) t = 0 (1) ( v)+ ( vv) t = p g (2) E t + { (E + p)v} = gv z (3) where the energy equation and the equation of state for an ideal gas are defined as: p State: E = ( 1) + 1 (v v) (4) 2 where is density, is pressure, is the fluid velocity, along with energy density

19 Contents EP711 Supplementary Material Thursday, September 4, 2014 Lagrangian vs. Eulerian Coordinates Advection and Conservation Conserved Physical Quantities Wave Equations

20 Wave Equations The hyperbolic PDEs that we will investigate in this course yield wave equation solutions when manipulated in the appropriate 2 2 c 2 r 2 u 2 2 c 2 =0 The wave equation can be described instead in terms of two advection equations with + =0 A system involving two advection equations and two variables can be constructed...

21 Idealized System of Two Advection Equations A system of two coupled PDEs describing advection of quantities u and v can @t + =0 Combined, they form a wave equations with characteristic speed c. This form is consistent with that describing onedimensional EM wave propagation, acoustic wave propagation, or wave propagation on a string.

22 Recall from Undergrad Maxwell s Equations: Note: Potter demonstrates calculation of Lax method stability for a linear system of PDEs based on Maxwell s equations! The same can be applied for acoustics (hint-hint)...

23 Linear Acoustics Equations The equations of linear acoustics describe conservation of mass density (continuity) and conservation of linear momentum, which in the following system have been coupled via an ideal gas equation + @t + =0 Here, p is pressure, u is velocity, rho0 is density, and gamma is the ratio of specific heats.

24 Wave Equation Solution Like general wave equation solutions, the wave equation can be obtained as =0 Substitute from momentum 2 2 p 0 2 = The speed of sound can be expressed: c 2 s = p 2 2 = c

25 Dispersion Relations Wave dispersion relations relate the effective wavenumber and frequency of waves within a system under the assumption of Fourier time-harmonic solutions of the form: u(x, t) =U o e j(!t 2 2 c 2 =0! 2 + c 2 k 2 =0 The resulting expression can be rewritten as:! = ck Which is an ideal dispersion (non-dispersive!) relation for waves with phase velocity and group velocity equal to c. NOTE: c is not the fluid velocity, but a characteristic speed of the physical system