The Euler equations in fluid mechanics

Transcription

1 The Euler equations in fluid mechanics Jordan Bell Department of Mathematics, niversity of Toronto April 14, Continuity equation Let Ω be a domain in R n and let C (Ω R); perhaps later we will care about functions that are in larger spaces, and to justify making conclusions about those we will have to check that what we have said here applies to them. Let be a Lipschitz domain in Ω. Thinking of as a density, the amount of stuff in at time t is (x, t)dx. Let q C (Ω R) and F C (Ω R, R n ). We think about q(x, t) as the rate at which new stuff appears at point x at time t, and F as the flux of the stuff. A change in the total amount of stuff in occurs from stuff appearing inside and from stuff going through the boundary of. We formalize this as the statement d dt (x, t)dx = q(x, t)dx F (s, t) N(s)ds, where N(s) is the outward pointing unit normal to the surface at the point s. sing the divergence theorem we get F (s, t) N(s)ds = (div F )(x, t)dx, and hence or, ( t )(x, t) = d dt (x, t)dx = ( t q + div F )(x, t)dx. (q(x, t) (div F )(x, t))dx, Because this is true for any Lipschitz domain in Ω, it follows that the integrand is 0: for all x Ω and t R, we have ( t q + div F )(x, t) = 0, 1

2 i.e. t + div F = q. This is called a continuity equation. If (x, t) denotes the density of stuff at the point x at time t and u denotes the velocity of the stuff at the point x and time t, then the flux F (in other words, the momentum), is F = u. If there is no stuff spontaneously appearing, but rather stuff only moves around, then q = 0, and so t + div (u) = 0. (1) One can describe the statement that stuff is not spontaneously appearing as conservation of mass, and hence (1) can be thought of as a consequence of conservation of mass. Momentum The integral (u)(x, t)dx is the total amount of momentum of the stuff at points in at time t. We postulate that there is a function p C (Ω, R), which we call pressure, such that the rate of change of the total amount of momentum over a set at time t is equal to the flow of momentum from outside to inside the set at time t plus the total amount of inward directed pressure over the boundary of the set at time t, which here means d dt (u)(x, t)dx = (u)(s, t)u(s, t) N(s)ds p(s, t)n(s)ds, where N(s) is the outward pointing unit normal to the surface at s. sing the divergence theorem, d (u)(x, t)dx = div (u u)(x, t) ( p)(x, t)dx. dt Combined with this gives d (u)(x, t)dx = t (u)(x, t)dx, dt ( t (u) + div (u u) + p)(x, t)dx. Because this is true for any Lipschitz domain in Ω, we obtain for all x Ω and t R, or ( t (u) + div (u u) + p)(x, t) = 0 t (u) + div (u u) + p = 0. ()

3 To state that the stuff we are talking about is incompressible means that is constant. For the rest of this note, unless we state otherwise we take to be a nonzero constant, with which equation (1) becomes and () becomes div (u) = 0, (3) t u + div (u u) + 1 p = 0. (4) The two equations (3) and (4) are called the Euler equations for an incompressible fluid. Taking the divergence of (4) yields t div (u) + div (div (u u)) + 1 div ( p) = 0. sing (3) and writing p = div ( p), div (div (u u)) + 1 p = 0. (5) As div (u u) = j (u i u j )e i, we have, using (3), div (div (u u)) = i j (u i u j ) Therefore, using this with (5) we get = i (( j u i )u j + u i j u j ) = i (( j u i )u j + u i div (u)) = i (( j u i )u j ) = ( i j u i )u j + ( j u i ) i u j = ( j (div (u)))u j + ( j u i ) i u j = ( j u i ) i u j. p = ( j u i ) i u j. (6) The use of this equation is to give us more information about the pressure p. Furthermore, with u = u i e i and writing u = ( j u) e j = ( j (u i e i )) e j = j u i e i e j, the contraction of the tensor u with itself is ( u)( u) = ( j u i e i e j )( l u k e k e l ) = ( j u i )( l u k )(e i e j )(e k e l ) = ( j u i )( l u k )δ j,k e i e l = ( j u i )( l u j )e i e l, 3

4 for which With this, equation (6) becomes As Tr (( u)( u)) = ( j u i )( i u j ). p = Tr (( u)( u)). div (u u) = j (u i u j )e i = ( j u i )u j e i + u i j u j e i = ( j u i )u j e i + u i div (u)e i, using (3) we have and hence it follows from (3) that div (u u) = ( j u i )u j e i, div (u u) = u u. (7) This expression for div (u u) may be easier to work with than the original expression. 3 Energy If v = v i e i is a vector field, we write Then, v = ( j v) e j = ( j v i )e i e j. v v = (v k e k ) (( j v i )e i e j ) = v j j v i e i = v j j v. If u (velocity of stuff) and p (pressure of stuff) satisfy (3) and (4), then applying u to both sides of (4) we get First, u ( t u) + u div (u u) + 1 u p = 0. (8) t (u u) = t (u i u i ) = ( t u i )u i + u i ( t u i ) = u i ( t u i ) = u ( t u). Second, so but div (u u) = j (u i u j )e i u div (u u) = u i j (u i u j ) = u i ( j u i )u j + u i u i j u j ; div ((u u)u) = div (u i u i u j e j ) = j (u i u i u j ) = u i ( j u i )u j + u i u i j u j, hence div ((u u)u) = u div (u u) u udiv (u), 4

5 and using (3) this is div ((u u)u) = u div (u u). Third, div (pu) = ( p) u + pdiv (u), and using (3) this is div (pu) = ( p) u. Putting these three results into (8) gives or We define 1 t(u u) + 1 div ((u u)u) + 1 div (pu) = 0, t ( 1 u u ) + div ( ) 1 (u u)u + pu = 0. (9) E = 1 u u. If is thought of as mass density, with units of kg/m 3, and u is thought of as the velocity of stuff, with units of m/s, then E has units of kg m 1 s = J/m 3. We choose to think of E defined this way as energy density; we say choose because although E has the right units to be energy density, any multiple would have the same units, and it is not apparent from what we have said so far why we care about 1 u u rather than some other multiple of u u. Writing equation (9) using E gives t E + div (Eu + pu) = 0, which is thus a statement about the rate of change of energy density. We call E+p the total specific enthalpy of the stuff. To say that a quantity is specific means that it expresses some quantity per kg, and the dimensions of enthalpy are J. 4 Vorticity In this section, unless we say otherwise we take n = 3. For vector fields v, w, (v w) = v w + w v + v curl w + w curl v. sing this identity v = u and w = u gives and therefore (7) can be written as u u = u curl u 1 (u u), div (u u) = 1 (u u) u curl u. (10) 5

6 Taking the curl of (4) yields t curl u + curl div (u u) = 0; we used the fact that the curl of the gradient of any scalar field is 0 and so curl p = 0. sing (10), this becomes ( ) 1 t curl (u) + curl (u u) u curl u = 0, and as the curl of the gradient of a scalar field is 0, this is For vector fields v, w, t curl u = curl (u curl u). curl (v w) = vdiv w wdiv v + ( v)(w) ( w)(v), and with v = u and w = curl u we obtain t curl u = udiv curl u curl (u)div u + ( u)(curl u) ( curl u)(u). Because the divergence of the curl of a vector field is 0 and because div u = 0 by (3), this becomes t curl u = ( u)(curl u) ( curl u)(u). We call ω = curl (u) the vorticity of the stuff, and with this notation the above equation can be written as 5 Material time derivative t ω = ( u)(ω) ( ω)(u). (11) One often deals with expressions like t ω + u ω, and we write D Dt = t + u, D and call Dt the material time derivative; it depends on the velocity u of the stuff. With this notation, the equation (11) is Dω Dt = ω u. sing (7) (which itself supposes (3)), we can write (4) using the material time derivative as Du + p = 0. Dt 6

7 6 Irrotational velocity fields In this section unless we say otherwise we take n = 3 and we suppose that curl u = 0, which we describe as u being irrotational. We suppose also in this section that Ω is simply connected, which together with curl u = 0 implies that there is some φ C (Ω R) for which u(x, t) = ( φ)(x, t) for all x Ω and for all t R; cf. the Helmholtz decomposition of a vector field in R 3. We call φ a potential function for u. Combining (4), (7), and u = φ, we obtain t φ + ( φ) φ + 1 p = 0. (1) We have ( φ) φ = ( i φe i ) ( k φe k ) = ( i φe i ) ( j k φe k e j ) = ( i φ) i k φe k = 1 k(( i φ)( i φ))e k with which (1) becomes = 1 k( φ φ)e k = 1 ( φ φ), t φ + 1 ( φ φ) + 1 p = 0, or ( t φ + 1 φ φ + 1 ) p = 0. Then, defining P to be P = t φ + 1 φ φ + 1 p, we have that P depends only on time. We call P the total pressure, and the statement that the total pressure depends only on time if the velocity u is irrotational is called Bernoulli s principle. Furthermore, combining (3) with u = φ gives φ = 0, i.e., for each t R, x φ(x, t) is a harmonic function on Ω. 7

8 7 Euler equations in one dimension In this section we take n = 1 and do not suppose that the pressure is constant. Since we do not take to be constant, we will use (1), which tells us that and (), which tells us that t + div (u) = 0, t (u) + div (u u) + p = 0. As n = 1 here, we can write these two equations as and t + x (u) = 0 (13) t (u) + x (u ) + x p = 0. (14) We suppose, giving no justification, that there are some constant K and γ for which p = K γ. With this assumption, equation (14) becomes t (u) + x (u ) + γ p x = 0. (15) We write = where 0 is a constant, and we also write u = u 0 + u 1 where u 0 is a constant. With these definitions, the equation (13) becomes i.e. t ( ) + x ( 0 u u u u 1 ) = 0, t x u 1 + u 0 x 1 + x ( 1 u 1 ) = 0. Supposing that the last term is negligible, an approximation to the above equation is t x u 1 + u 0 x 1 = 0. (16) Furthermore, (15) becomes t ( 0 u u u u 1 ) + x ( 0 u u 0 u u u u 0 u u 1) p + γ x ( ) =0. sing that 0 and u 0 are constant and supposing that x ( 1 u 1 ) and x (u 1) are negligible gives us the approximation 0 t u 1 + u 0 t u 0 x u 1 + u p 0 x 1 + γ x

9 1 Expressing 0+ 1 as a geometric series in powers of 1 0 and supposing that the sum of all the nonconstant terms is negligible, and approximating p = K γ as p 0 = K γ 0 gives us the approximation 0 t u 1 + u 0 t u 0 x u 1 + u 0 x 1 + γ p 0 0 x 1 = 0. Combining this equation with (16) multiplied by u 0 yields 0 t u u 0 x u 1 + γ p 0 0 x 1 = 0. We define D t = t + u 0 x, with which we can write the above equation as 0 D t u 1 + γ p 0 0 x 1 = 0, (17) and we can write (16) as Applying D t to (18) gives D t x u 1 = 0. (18) D t x D t u 1 = 0, and then using (17) this becomes ( Dt 1 + x γ p ) 0 x 1 0 or D t 1 γ p 0 0 x 1 = 0, which is a wave equation satisfied by 1. = 0, 9