CHAPTER 6 Derive Differential Continuity, Momentum and Energy Equations Form Integral Equations for Control Volumes

CHAPTER 6 Derive differential Continuity, Momentum and Energy equations form Integral equations for control volumes.

Simplify these equations for 2-D steady, isentropic flow with variable density

CHAPTER 8 Write the 2 –D equations in terms of velocity potential reducing the three equations of continuity, momentum and energy to one equation with one dependent variable, the velocity potential.

CHAPTER 11 Method of Characteristics exact solution to the 2-D velocity potential equation. Gauss's Theorem - Divergence Theorem transforms a surface integral into a volume integral

r r r ∫∫(V)dS = ∫∫∫(∇V)d vol where : (V) is a vector S vol

∫∫(a)dS = ∫∫∫(∇a)d vol where : ( a) is a scalar S vol ∂( ) ∂( ) ∂( ) Gradient ∇ = i + j+ k ∂x ∂y ∂z ∇ of a vector is a scalar ∇ of a scalar is a vector CONTINUITY EQUATION CONSERVATIVE INTEGRAL FORM

V, velocity vector

dS control volume open thermodynamic system region in space

− ∫∫ρ V dS net of mass leaving the control volume. S (by convention mass inflow is +) ∂ ∫∫∫ρ d vol change in mass inside the control volume ∂t vol ∂ ∫∫()ρ V dS = ∫∫∫ρ d vol Continuity Equation in integral (conservative) form S ∂t vol CONTINUITY EQUATION CONSERVATIVE INTEGRAL FORM

Gauss's Theorm transforms a surface integral into a volume integral r r ∂ ∂ ∂ ∫∫VdS = ∫∫∫∇Vd vol where, ∇ = + + S vol ∂x ∂y ∂z ∆ (control volume mass) = net mass outflow ∂ r ∫∫∫ρ d vol = − ∫∫()ρV dS ∂t vol S by convention mass inflow is +. applying Gauss's Theorm to the net mass outflow term, CONTINUITY EQUATION CONSERVATIVE INTEGRAL FORM ∂ r ∫∫∫ ρ d vol = −∫∫∫∇(ρV) d vol vol ∂t vol ∂ρ r + ∇(ρV) = 0 (6.50) ∂t unsteady, 3- D, any fluid, variable density ∂ρ ∂()ρ u ∂()ρ v ∂()ρ w + + = 0 ∂t ∂x ∂y ∂z ∂()ρ u ∂ρ ∂u substituting, = u + ρ in x, y and z ∂x ∂x ∂x ∂ρ  ∂ρ ∂ρ ∂ρ   ∂u ∂v ∂w  + u + v + w  + ρ + ρ + ρ  = 0 ∂t  ∂x ∂y ∂z   ∂x ∂y ∂z  MONENTUM EQUATION CONSERVATIVE INTEGRAL FORM d()mV F = = change in momentum V, velocity vector dt Forces dS Body Force ∫∫∫ρ f d vol, vol where f is the body force constant Pressure Force − p dS control volume ∫∫ open thermodynamic system S region in space Viscous Force ∫∫τ dS S Momentum change inside the volume ∫∫()ρ V dS V S ∂()ρ V Change of Momentum with time ∫∫∫ d vol vol ∂t MomentumChange = Body Force + Pressure Force + Viscous Force ∂()ρ V ∫∫()ρ V dS V + ∫∫∫ d vol = ∫∫∫ρ f d vol − ∫∫p dS + ∫∫τ dS S vol ∂t vol S S MONENTUM EQUATION CONSERVATIVE INTEGRAL FORM

∂(ρ V) ∫∫()ρ V dS V + ∫∫∫ d vol= ∫∫∫ρ f d vol − ∫∫p dS+ ∫∫τ dS S vol ∂t vol S S using Gauss' s Therom (6.1), ∫∫AdS = ∫∫∫()∇A d vol and ∫∫(a)dS= ∫∫∫(∇a)d vol S vol S vol to convert the three surface integrals to volume inteagrals. ∂()ρ V ∫∫∫∇()ρ V Vd vol+ ∫∫∫ d vol= ∫∫∫ρ f d vol− ∫∫∫∇p d vol+ ∫∫∫∇τ d vol vol vol ∂t vol vol vol differentiating,

∂()ρ V = −∇p −∇()ρ V V −∇τ+ρ f ∂t MOMENTUM EQUATIONS unsteady, 3D, any fluid, variable density

∂(ρ V) = −∇p − ∇(ρ V)V − ∇τ + ρ f ∂t

∂ ∂p  ∂u ∂u ∂u   ∂ ∂ ∂  ρu = − − ρu + ρv + ρw  −  τxx + τyx + τzx  + ρfx ∂t ∂x  ∂x ∂y ∂z   ∂x ∂y ∂z  ∂ ∂p  ∂v ∂v ∂v   ∂ ∂ ∂  ρv = − − ρu + ρv + ρw  −  τxy + τyy + τzy  + ρf y ∂t ∂y  ∂x ∂y ∂z   ∂x ∂y ∂z  ∂ ∂p  ∂w ∂v ∂w   ∂ ∂ ∂  ρw = − − ρu + ρv + ρw  −  τxz + τxz + τzz  + ρfz ∂t ∂z  ∂x ∂y ∂z   ∂x ∂y ∂z  restricting the momentum equation to Newtonian fluids for which the fluids stress is a linear function of the rate of deformation of the fluid - the change of velocity with distance. du for 1 D, τ = µ dx

∂u 2  ∂u ∂v  τ = −2µ + µ()∇V τxy = τ yx = −µ +  xx ∂x 3  ∂y ∂x  ∂v 2 τ = −2µ + µ()∇V  ∂v ∂w  yy τyz = τzy = −µ +  ∂x 3  ∂z ∂y  ∂w 2 τzz = −2µ + µ()∇V  ∂w ∂u  ∂x 3 τxy = τyx = −µ +   ∂x ∂z  ENERGY EQUATION CONSERVATIVE INTEGRAL FORTM

First Law Q = ∆E + W = ∆E + Wshaft + Wviscous + Wpressure + Wbody Work = Force×Velocity

Wshaft = 0 Work − ()p dS V pressure ∫∫ S Work (ρ f d vol)V body ∫∫∫ Vol Work − ()τ dS V viscous ∫∫ S  V2  Net Energy into control volume ∫∫()ρ V dS e +  S  2  ∂  V2  Change in energy inside the control volume ∫∫∫ρe +  d vol ∂t Vol  2  Heat addition ∫∫qdS S

Internal energy, U = cvT First Law Q = ∆E + W = ∆E + Wshaft + Wviscous + Wpressure + Wbody

Q = ∆Enet in + ∆Echange in Wshaft + Wviscous + Wpressure + Wbody control control volume volume  V2  ∂  V2  Q = ∫∫()ρ ∇ dS e +  + ∫∫∫ρe + d vol + −∫∫()τ dS V − ∫∫()p V dS + ∫∫∫(ρ f d vol)V = (2.20a) S  2  ∂t vol  2  S S vol

∂  V2    V2  ρc T +  = −∇ρ ∇c T +  − ∇ • q − ∇ • pV − ∇ • (τ• V) + ρ(g • V)  v    v  ∂t  2    2  ∂  V2   ∂  V2  ∂  V2  ∂  V2  ρc T +  = − uρc T +  + wρc T +  + wρc T +   v    v   v   v  ∂t  2   ∂x  2  ∂y  2  ∂y  2 

 ∂q x ∂q y ∂qz   ∂ ∂ ∂  −  + +  −  ρu + ρv + ρw   ∂x ∂y ∂z   ∂x ∂y ∂z   ∂ ∂ ∂  −  ()τxxu + τxyv + τxz w + ()τyxu + τyyv + τyzw + ()τzxu + τzyv + τzz w   ∂x ∂y ∂z 

 ∂T ∂T ∂T ∂T   ∂q ∂q y ∂q   ∂p   ∂u ∂v ∂w   x z  ρcv  + u + v + w  = − + +  − T   + +   ∂t ∂x ∂y ∂z   ∂x ∂y ∂z   ∂T p  ∂x ∂y ∂z   ∂u ∂v ∂w    ∂u ∂v   ∂u ∂w   ∂v ∂w  − τ + τ + τ  − τ  +  + τ + + τ  +   xx yy zz   xy   xz   yz    ∂x ∂y ∂z    ∂y ∂x   ∂z ∂x   ∂z ∂y  EQUATION SUMMARY - 3D, viscous, variable density CONTINUITY ∂ρ  ∂ρ ∂ρ ∂ρ   ∂u ∂v ∂w  + u + v + w  + ρ + ρ + ρ  = 0 ∂t  ∂x ∂y ∂z   ∂x ∂y ∂z 

MOMENTUM − x, y,z directions ∂ ∂p  ∂u ∂u ∂u   ∂ ∂ ∂  ρu = − − ρu + ρv + ρw  −  τxx + τyx + τzx  + ρfx ∂t ∂x  ∂x ∂y ∂z   ∂x ∂y ∂z  ∂ ∂p  ∂v ∂v ∂v   ∂ ∂ ∂  ρv = − − ρu + ρv + ρw  −  τxy + τyy + τzy  + ρf y ∂t ∂y  ∂x ∂y ∂z   ∂x ∂y ∂z  ∂ ∂p  ∂w ∂v ∂w   ∂ ∂ ∂  ρw = − − ρu + ρv + ρw  −  τxz + τxz + τzz  + ρfz ∂t ∂z  ∂x ∂y ∂z   ∂x ∂y ∂z 

ENERGY

2D steady incompressible, inviscid MOMENTUM − x, y,z directions ∂ ∂p  ∂u ∂u ∂u   ∂ ∂ ∂  ρu = − − ρu + ρv + ρw  −  τxx + τyx + τzx  + ρfx ∂t ∂x  ∂x ∂y ∂z   ∂x ∂y ∂z  ∂ ∂p  ∂v ∂v ∂v   ∂ ∂ ∂  ρv = − − ρu + ρv + ρw  −  τxy + τyy + τzy  + ρf y ∂t ∂y  ∂x ∂y ∂z   ∂x ∂y ∂z  ∂ ∂p  ∂w ∂v ∂w   ∂ ∂ ∂  ρw = − − ρu + ρv + ρw  −  τxz + τxz + τzz  + ρfz ∂t ∂z  ∂x ∂y ∂z   ∂x ∂y ∂z 

ENERGY

Divide a flow into two regions according to the forces that prevail

BOUNDARY LAYER FREE STREAM thin layer near wall τ = 0, µ = 0, viscous forces as improtant as interial forces Potential Flow ∂u ∂u large, τ = µ very large isentropic, frictionless ∂y ∂y irrotational, ignore traverse momentum equations uniform and parallel 2 − D incompresible boundary layer equations, ∂u ∂u 1 dp 1 ∂τ u + v = − + yx ∂y ∂y ρ dx ρ dy ∂u ∂u 1 dp µ ∂ 2u u + v = − + ∂y ∂y ρ dx ρ dy2 ∂u ∂v + v = 0 ∂x ∂y EQUATION SUMMARY - 3D, viscous, variable density CONTINUITY ∂ρ  ∂ρ ∂ρ ∂ρ   ∂u ∂v ∂w  + u + v + w  + ρ + ρ + ρ  = 0 ∂t  ∂x ∂y ∂z   ∂x ∂y ∂z 

ENERGY

 ∂T ∂T ∂T ∂T   ∂q ∂q y ∂q   ∂p   ∂u ∂v ∂w   x z  ρcv  + u + v + w  = − + +  − T   + +   ∂t ∂x ∂y ∂z   ∂x ∂y ∂z   ∂T p  ∂x ∂y ∂z   ∂u ∂v ∂w    ∂u ∂v   ∂u ∂w   ∂v ∂w  − τ + τ + τ  − τ  +  + τ + + τ  +   xx yy zz   xy   xz   yz    ∂x ∂y ∂z    ∂y ∂x   ∂z ∂x   ∂z ∂y  2-D, steady, inviscid (isentropic), variable density

CONTINUITY  dρ dρ   du dv  ∂( ) u + v  + ρ + ρ  = 0 = 0  dx dy   dx dy  ∂t ∂( ) = 0 MOMENTUM − x, y,z directions ∂z ∂p  ∂u ∂u  = −ρu + ρv  w = 0 ∂x  ∂x ∂y  τ = 0 ∂p  ∂v ∂v  = −ρu + ρv  ∂y  ∂x ∂y 

ENERGY

 ∂T ∂T   ∂q ∂q y   ∂p   ∂u ∂v   x  ρcv u + v  = − +  − T   +   ∂x ∂y   ∂x ∂y   ∂T p  ∂x ∂y  VELOCITY POTENTIAL – reduce to one equation

u and v are functions Vdl = 0 for irrotational flow ∫ of the same scalar qauntity, C isentropic, τ = 0, µ = 0 define as Φ, velocity potential function ∂()Φ ∂()Φ Vdl is independent of path u = , v = ∂x ∂y an exact differential, CHECK : Greens Theorem, → dependent only on position ∫ ∫∫ C S ∂() ∂()  ∂v ∂v  exact differential d( ) = dx + dy V dl =  − dxdy = 0 ∂x ∂y ∫ ∫∫ C S  ∂x ∂x  ∂()Vdl ∂()Vdl ∂v ∂v d(Vdl) = dx + dy = ∂x ∂y ∂x ∂x ∂()Φ ∂()Φ d(Vdl) = u dx i + u dy j substituting, u = , v = ∂x ∂y ∂()Vdl ∂()Vdl ∂ 2 ()Φ ∂2 ()Φ by comparison, u = , v = = ∂x ∂y ∂xdy ∂ydx CONTINUITY EQUATION 2 - D steady, inviscid, varable density

 dρ dρ   du dv  u + v  + ρ + ρ  = 0  dx dy   dx dy  continuty equation in terms of velocity potential ∂()Φ du ∂ 2 ()Φ substitute : u = = Φ , = = Φ ∂x x dx ∂x 2 xx ∂()Φ dv ∂ 2 ()Φ v = = Φ , = = Φ ∂y x dy ∂x 2 xx ∂()Φ dρ ∂()Φ dρ ∂ 2 ()Φ ∂ 2 ()Φ + + ρ + ρ = 0 ∂x dx ∂y dy ∂x 2 ∂x 2 dρ dρ 2 variables, ρ and Φ Φ + Φ + ρΦ + ρΦ = 0 x dx x dy xx yy density will be eliminated by the momentum equations. substitute : MOMENTUM EQUATIONS ∂()Φ u = = Φ multipy x direction by dx ∂x x ∂p ∂u ∂u du ∂2 ()Φ − dx = ρu dx + ρv dx = = Φ ∂x ∂x ∂y dx ∂x 2 xx since for irrotational flow, ∂()Φ v = = Φy du dv ∂y = 2 dy dx dv ∂ ()Φ = 2 = Φyx ∂p ∂u ∂v dy ∂x − dx = ρu dx + ρv dx ∂p ∂x ∂x ∂x − dx = ρ()Φ Φ + Φ Φ ∂x x xx y yx for the y direction equation, ∂p − dy = ρ()Φ Φ + Φ Φ ∂y x xy y yy 2  ∂p  a =    ∂ρ S  ∂p  ∂ρ =    a 2  ∂ρ 1 ∂p = ∂x a 2 ∂x ∂ρ ρ − = ()Φ Φ + Φ Φ ∂x a 2 x xx y yx ∂ρ ρ − = ()Φ Φ + Φ Φ ∂y a 2 x xy y yy substituting into the contuinuity equation, ∂ρ ρ − = ()Φ Φ + Φ Φ ∂x a 2 x xx y yx ∂ρ ρ − = ()Φ Φ + Φ Φ ∂y a 2 x xy y yy  Φ2   Φ2  Φ Φ 1− x Φ + 1− y Φ − 2 x y (8.17, for 2 − D)  2  xx  2  yy 2  a   a  a  Φ2   Φ2  Φ Φ 1− x Φ + 1− y Φ − 2 x y Φ (8.17,for 2 − D)  2  xx  2  yy 2 yy  a   a  a substituting, u = Φx , v = Φ y u 2  v2  uv (1− )Φ + 1− Φ − 2 Φ = 0 (11.5) 2 xx  2  xy 2 yy a  a  c ∂Φ ∂Φ exact differentials for and , dx dy  ∂Φ  ∂2Φ ∂2Φ d  = dx + dy = Φ xxdx + Φ xydy = du (11.6)  dx  ∂x 2 ∂x∂y  ∂Φ  ∂2Φ ∂2Φ d  = dx + dy = Φ xydx + Φ yydy = dv (11.7)  dy  ∂xdy ∂y∂y 3 simultaneous linear equations in Φ u 2 uv  v2  (1− )Φ − 2 Φ + 1− Φ = 0 2 xx 2 xy  2  yy a c  a 

Φxxdx + Φxydy + 0 = du

Φxydx + 0 + Φyydy = dv u 2 v2 (1− 2 ) 0 (1− 2 ) a a Φ is indeterminate, dx du 0 xy on a characteristic of the solution, 0 dv dy When both N and D are 0 N Φ = = Φxy is indeterminate. xy u 2 uv v2 D (1− ) - 2 (1− ) N = 0 defines the characteristic a 2 a 2 a 2 of the solution, C = f(x, y). dx du 0 D = 0 defines properties along 0 dv dy the charactersitic. N,numerator = 0 D,denominator = 0 2 2 u v 2 2 (1− ) dudy + (1− )dxdv = 0 u 2 uv v 2 a 2 a 2 (1− )()dy + 2 dxdy+ (1− )(dx) = 0 a2 a2 a2 2 2 2 u  uv u + v  2 2 2  - ± −1  u  dy  uv  dy  v (1− 2 ) 2 2 (1− ) + 2 + (1− ) = 0 du  a a  2   2   2 = a a  dx  a  dx  a dv v2  u 2  (1− )  1−  2 2 2  2  uv u + v a  a  − ± −1  dy  a2 a2   = 2 dV 2 dθ = ± M −1  dx characteristic  u  1−  V  2   a  dθ is the Prandtl - Meyer Function ∫ uv − ± ()M2 −1 −1  dy  2 = a   2 θ + υ(M) = K alongC characteristic  dx characteristic  u  − − 1−   a2  θ − υ(M) = K +alongC+characteristic  