Conservation of Mass

Conservation of Mass (x , y ,z ) We consider a very small 0 0 0 volume element of air δV = δxδyδz that is centered at the point x , y ,z ()0 0 0 δz Conserv ation of mass r equ ir es δy that the local rate of change of δx mass must equal the net rate of mass inflow per unit volume. Rate of inflow of mass through left wall: MLx Rate of outflow of mass through right wall: MRx (x0 , y0 ,z0 ) Net inflow of mass into the volume (x-direction only): Mx = MLx -MRx MLx MRx δz Method: Use Taylor expansion to develop mathematical expressions δy for the inflow of mass through the walls of this fluid element. δx 1 Taylor series expansion: f ()x = f ()x0 + f ′ ()(x0 x − x0 )+ higher order terms (x0 , y0 ,z0 ) In this case (neglecting higher order terms): ∂ ρu()x = ρu (x )+ ()()ρu x − x 0 ∂x 0 MLx MRx Therefore, we can express the δz rates of inflow and outflow per unit area as: δy ⎛ ∂ δx ⎞ M Lx = ⎜ ρu − ()ρu ⎟δyδz ⎝ ∂x 2 ⎠ δx ⎛ ∂ δx ⎞ M Rx = ⎜ ρu + ()ρu ⎟δyδz ⎝ ∂x 2 ⎠ Since the area of the left and right faces of the volume is δyδz, ⎡ ∂ δx⎤ ⎡ ∂ δx⎤ ∂ ρu − ()ρu δyδz − ρu + ()ρu δyδz = − ()ρu δxδyδz ⎣⎢ ∂x 2 ⎦⎥ ⎣⎢ ∂x 2 ⎦⎥ ∂x is the net rate of mass flow into the volume (x0 , y0 ,z0 ) due to the x velocity component. We can derive similar expressions for the net rate of mass flow due to the M M y and z velocity components. Lx Rx Summing over all three components δz yields the net rate of mass inflow: ⎡ ∂ ∂ ∂ ⎤ δy − ⎢ ()ρu + ()ρv + ()ρw ⎥δxδyδz ⎣∂x ∂y ∂z ⎦ δx 2 ⎡ ∂ ∂ ∂ ⎤ −⎢ ()ρu + ()ρv + ()ρw ⎥δxδyδz = net rate of mass inflow ⎣∂x ∂y ∂z ⎦ r −∇ ⋅(ρV ) = net rate of mass inflow per unit volume The net rate of mass inflow per unit volume must equal the rate of mass increase per unit volume, which is the local rate of change of density. r ∂ρ −∇ ⋅(ρV )= ∂t ∂ρ r Mass divergence + ∇ ⋅(ρV )= 0 form of the ∂t continuity equation local rate divergence of change of mass of density flux ∂ρ r To derive an alternative form of the continuity + ∇ ⋅(ρV )= 0 equation, start with the mass divergence form: ∂t r r r Now apply the vector identity: ∇⋅(ρV )= ρ∇ ⋅V +V ⋅∇ρ ∂ρ r r + ρ∇ ⋅V +V ⋅∇ρ = 0 ∂t d ∂ r Next apply Euler’s relationship: ≡ +V ⋅∇ dt ∂t 1 dρ r + ∇ ⋅V = 0 ρ dt Velocity divergence form of the fractional rate of change of velocity density following the fluid motion divergence continuity equation 3 Scale Analysis of Continuity Equation When scaling the continuity equation, it is important to recognize that large variations in density are only relevant in the vertical. So we reppyresent the density field as the sum of a horizontal reference value and a deviation from that value: ρ()x, y, z,t = ρ0 (z)+ ρ′(x, y, z,t) Substitute into velocity divergence form of the continuity equation: 1 dρ r + ∇ ⋅V = 0 ρ dt 1 ⎡ ∂ r ⎤ r ′ ′ ⎢ ()()ρ0 + ρ +V ⋅∇ ρ0 + ρ ⎥ + ∇ ⋅V = 0 ρ0 ⎣∂t ⎦ 1 ⎡⎛ ∂ρ0 ∂ρ′ ⎞ ⎛ ∂ρ0 ∂ρ′ ⎞ ⎛ ∂ρ0 ∂ρ′ ⎞ ⎛ ∂ρ0 ∂ρ′ ⎞⎤ r ⎢⎜ + ⎟ + u⎜ + ⎟ + v⎜ + ⎟ + w⎜ + ⎟⎥ + ∇ ⋅V = 0 ρ0 ⎣⎝ ∂t ∂t ⎠ ⎝ ∂x ∂x ⎠ ⎝ ∂y ∂y ⎠ ⎝ ∂z ∂z ⎠⎦ 1 ⎡∂ρ′ ∂ρ′ ∂ρ′ dρ0 ∂ρ′⎤ r ⎢ + u + v + w + w ⎥ + ∇ ⋅V = 0 ρ0 ⎣ ∂t ∂x ∂y dz ∂z ⎦ 1 ⎡∂ρ′ r ⎤ w dρ0 r ⎢ +V ⋅∇ρ′⎥ + + ∇ ⋅V = 0 ρ0 ⎣ ∂t ⎦ ρ0 dz A BC ∂u ∂v ∂w d ρ′ U −7 −1 + + + w ()lnρ = 0 A: =10 s ∂x ∂y ∂z dz 0 ρ0 L If only the first three terms in the W −6 −1 B: =10 s above equation were present, then the H atmosphere would be incompressible. U W The above result implies that the C: 10−1 , =10−6 s−1 L H atmosphere behaves as if it were incompressible for purely horizontal motions (i.e., when w = 0). 4 Incompressible Fluids 1 dρ r If a fluid is incompressible, its density is + ∇ ⋅V = 0 constant following the fluid motion. Thus the ρ dt velocity divergence must be zero for an incompressible fluid. In the atmosphere, the compressibility associated with ∂u ∂v ∂w d + + + w ()lnρ0 = 0 the height dependence of ∂x ∂y ∂z dz density is important when there itiltiis vertical motion. Incompressibility is a very good approximation in the ocean, even when vertical motions are present. 5.

Load more