4.6 Selective withdrawl of thermally stratified fluid

Transcription

1 1 Lecture notes in Fluid Dynamics (1.63J/2.01J by Chiang C. Mei, MIT 4-6selw-therm.tex 4.6 Selective withdrawl of thermally stratified fluid [References]: R.C. Y. Koh, 1966 J. Fluid Mechanics, 24, pp Brooks, N. H., & Koh, R. C. Y., Selective withdrawal from density stratified reservoirs. J Hydraulics, ASCE, HY4, July Ivey, G. N. Monosmith, et. al. We now extend the analysis in the last section and consider the slow and steady flow of a thermally stratified fluid into a two- dimensional line sink. Thermal diffusion and convection now comes into play.

2 Governing equations We begin with the general law of mass conservation ρ + (ρ q = ρ + q ρ + ρ q = 0 (4.6.1 t t In environmental problems the range of temperature variation is within a few tens of degrees. The fluid density varies very little and obeys the following equation of state ρ = ρ o [1 β(t T o ] (4.6.2 where T denotes the temperature and β the coefficient of thermal expansion which is usually very small. Hence ( q ρ ρ ρ q = O 1 ρ and 1 ρ ρ t u ρ ρ 1 It follows that (4.6.1 is well approximated by q = 0 (4.6.3 which means that water is essentially incompressible. In two dimensions, we have Let Next, energy conservation requires that u x + w z = 0 (4.6.4 T t + q T = D 2 T (4.6.5 T = T + T (4.6.6 where T represents the static temperature when there is no motion, and T the motioninduced temperature variation. Therefore, T T o = ( T (z T o + T (x, z, t (4.6.7 and T t + q T + q T = D 2 T + D 2 T (4.6.8 The static temperature must satisfy 2 T = 0 (4.6.9

3 In a large lake with depth much smaller than the horizontal extent, the static temperature is essentially uniform horizontally. The Laplace equation reduces to D d2 T dt =0, implying dz2 dz The dynamic part is then gorvened by = constant ( T t + u T + w T z + w T z = D 2 T ( The exact equations for momentum balance are, in two dimensions, ρ ( w t + q w ρ ( u t + q u where p denotes the static part, which must satisfy = p + µ 2 u ( = p z p z gρ [ ] o 1 β(t + T T o + µ 2 w ( = p z gρ [ o 1 β(t To ] ( Taking the differenece of the two preceding equations, we find the equation for the dynamic part ρ ( w t + q w Approximation for slow and steady flow = p z + gρ oβt + µ 2 w ( For sufficiently slow flows, inertia terms can be ignored. Expecting that vertical motion is suppressed, we further assume that the vertical length scale δ is much smaller than the horizontal scale L, sothat / / z. The 2-D momentum equations can then be simplified to 0= p + u µ 2 ( z 2 0= p z + gβρ ot + µ 2 w z 2 ( Similarly we can linearize ( to get w d T dz = D 2 T ( z 2 Together (4.6.4, (4.6.18, ( and ( complete the lineaized governing equations. 3

4 4 Since Eliminating p from ( and (4.6.17, we get µ 2 z (u T 2 z w x =gβρ o ( w δ u = O 1, L w x u z = O ( 2 δ 1 L ( we can omit the second term on the left of ( In terms of the stream function defined by u = ψ z, w = ψ x ( ( becomes 4 ψ z 4 = gβρ o µ T ( Equation ( can be written as ψ x dt dz = D 2 T z 2 ( We now have just two equations for two unknowns ψ and T. The boundary conditions are or T u, w 0, as z ± ( ψ, ψ z T 0, as z ±. ( Let the volume rate of withdrawal be prescribed, we must then require the integal condition: udz = q, implying ψ(x, z = ψ(x, z = =q. ( Normalization Let ψ = qψ, T = T o T, x = Lx, z = δz ( Physically it is natural to choose the characteristic depth of thermal gradient as the global length scale L: ( L = β d T 1 ( dz

5 5 The scales T o and δ are yet to be specified. The dimensionless ( reads q δ 4 ( 4 ψ z 4 = gβρ ot o µl ( T, hence we choose so that q δ = gβρ ot o 4 µl ( ( 4 ( ψ T = ( z 4 Similarly, ( becomes T o δ 2 ( 2 T z 2 after normalization, suggesting the choice of + q d T DL dz ( ψ =0 T o δ = qdt/dz 2 DL ( so that ( 2 T + z 2 ( ψ =0 ( Eqs. ( and ( can be solved to give the scales δ = L1/3 α, where α = gβρ o d T dµd dz ( and The flux condition is normalized to T o = q ( δ 2 d T/dz DL ( ψ ( ψ ( = 1 (4.6.34

6 Similarity solution Let us try a one-parameter similarity transformation x = λ aˆx, z = λ b ẑ, ψ = λ c ˆψ, T = λ d ˆT ( The exponents a, b, c and d will be chosen so that the boundary value problem is formally the same as the original one To achieve invariance of (4.6.34, we set c = 0. In addition we set λ 4b = λ d a for (4.6.29, and λ d 2b = λ a for ( Hence, d a = 4b and a 2b = d implying b = d, a =3b = 3d. ( These relationships among the exponents suggest the following new similarity variables: ψ = f(ζ, T = h(ζ x 1/3 ( with ζ = z ( x 1/3 It is easily verified that these variables are invareiant under the similarity transformation. Carrying out the differentiations we get from ( ψ z = f x,ψ 1/3 zzzz = f x ( 4/3 T x = h 1 1 z x 1/3 3 x ( 4/3 1 1 = 3 ζh x + h 4/3 3 ( + h 1 x 4/ x 4/3 f = 1 3 (ζh + h ( Since T z = h 1 x, 2/3 ψ x = f z x 4/3 T zz = h 1 x = 1 3 f ζ 1 x ( 1 3

7 7 we get from ( or h 1 x f ζ 3 1 x =0 h ζ 3 f =0 ( The boundary conditions are transformed to f( f( = 1 ( and f,f,h 0 as ζ ± ( Mathematically, the similarity transformation has enabled us to reduce the boundary value problem involving partial differential equations to one with ordinary differential equations (4.6.39, (4.6.40, (4.6.41, and ( As long as x and z lie on the parabola z =constx 1/3, ψ and T x 1/3 are the same. From the transformation, we can also deduce that the boundary of the zone affected by the flow is a parabola, Along the centerline z = ζ = 0, the velocity varies as and the temperature varies as δ x 1/3 ( Umax ψ x x 1/3 ( Tmax x 1/3 ( The boundary value problem can now be solved by numerical means (such as Runge- Kutta. Numerical results by Koh (1966, Fig. 4 are shown in Figure ( In Koh (1966, stratification in fluid density is associated with the variation of concentration of a diffusive substance instead of temperature. The fluid density is governed by a diffusion equation formally the same as that for temperature here. To use his numerical results, f 0,h 0 in his plots are replaced by our f, h shown here. Extensive discussion on experimental confirmation as well as the three dimensional theory for a point sink can be found in Koh.

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