Incompressible Viscous Flows

Transcription

1 Incompessible Viscous Flows F an incompessible fluid, the continuity equation and the Navie-Stokes equation ae given as v = 0, () + v v = P + ν t v Using a vect identity, Equation () may be estated as v. () v + t v v ( v) = ν v (3) Define vticity ω = v (4) and taking cul of (3) we find ω t ( v ω) = ν ω Noting that cul of gadient is zeo and ( v ω) = ( ω) v ( v) ω + ω v v ω. (5) (6) Equation (5) may be estated as t ω + v ω = ω v + ν ω (7) Equation (7) is the vticity tanspt equation. It shows that in addition to being convected and diffused, vticity is also geneated by the fist on the ight hand side of Equation (7) by a vtex stetching mechanism. Two-Dimensional Plane Flows F two dimensional flows in xy-plane as shown in Figue, let

2 i j k ψ ψ v = ( kψ) = = i j, (8) x y z y x 0 0 ψ y v V u x Figue. Schematics of plane flows in a Catesian codinate system. That is ψ u =, y ψ v =, (9) x and Equation () is satisfied. The nonzeo element of ω is ω Z v u = ω = x y = ψ. (0) Equation (7) in two-dimensional case educes to ω ω ω 4 + u + v = ν ω. () t x y Using (0), Equation () may be estated as ψ ψ ψ + ψ ψ = ν 4 ψ. () t y x x y

3 ψ + t ( ψ, ψ) 4 ( x, y) = ν ψ. (3) Equation (3) is the equivalent to the Navie-Stokes equation and contains a single unknown ψ. Plane Flows in a Cylindical Geomety Case (a) v z = 0 and v and v ae functions of and. y V v v x Figue. Schematics of plane flows in a pola codinate system. That is, F a plane flow in cylindical geomety as shown in Figue, let ( ψ( ) ) v =,. (4) e z ψ ψ v = e e, (5) ψ v =, ψ v = (6) 3

4 The nonzeo element of ω is given by ωz = ω = ψ (7) whee ψ ψ ψ = + (8) Equation (7) now becomes ω + v t ω + v ω = ν ω. (9) Using (7), Equation (9) may be estated as ψ ψ ψ + ψ t 4 ψ = ν ψ, (0) ψ + t ( ψ, ψ) 4 (, ) = ν ψ. () Equation () is the equation govening ψ (,) codinated system. in plane flows expessed in pola Case (b) v = 0, v and v z ae functions of and z. z v z V v Figue 3. Schematics of axisymmetic flows in a cylindical codinate system. 4

5 F an axisymmetic flow in cylindical codinates, let ψ v = e (, z) () That is ψ ψ v = e + e z, (3) z v z ψ =, v ψ =. (4) z The vticity define by Equation (4) now becomes v v z ψ ψ ω = v = e = e + (5) z z Thus, the only nonzeo component of vticity is given by ω = ω = E ψ, (6) whee ψ ψ ψ E ψ = +. (7) z Equation (7) may now be estated as ω t + v ω + v ω z v ω ( ω ) comp. z = ν e (8) Using (6) in (8) we find t ψ ψ ψ ( E ψ) ( E ψ) + ( E ψ) + E ψ = νe ψ 4 z z z (9) 5

6 t ( ) ( E ψ, ψ) E ψ 4 (, z) + ψ E z ψ = νe ψ (30) Equation (30) govens (, z) ψ in axisymmetic cylindical flows. Spheical Codinates Spheical codinate system is shown in Figue 4. Hee x = coscosϕ y = cossin ϕ z = sin (3) z ϕ y x Figue 4. Schematics of spheical codinate system. Conside the case when v ϕ = 0 and v and v ae only functions of and. Let (, ) eϕψ v =, (3) sin That is 6

7 ψ ψ v = e e (33) sin sin = ψ sin v, ψ v =. (34) sin The vticity equation will educe to t ( ) ( E ψ, ψ) E ψ + 4 sin (, ) E ψ ψ ψ + cos sin = νe sin ψ, (35) whee cot E = +. (36) Intinsic Codinates It is sometimes simple to wk with a codinate system, which is attached to the suface of evolution. F the body of evolution shown Figue 5, conside the unit vects n, s, e ϕ. F this system the metics ae Thus h = h, h h =, h = h (37) n = Φ = = ϕ 3 s = Φ Φ Φ n + s + eϕ (38) n s ϕ The steam function f axisymmetic flows may now be intoduced as That is ( s,n) ψ ψ ψ v = eϕ = n + s (39) s n v n ψ =, s v s ψ = (40) n 7

8 8 z n e φ s φ Figue 5. Schematics of intinsic codinate systems. The vticity is now given as ψ = ψ + ψ = = ϕ E s s n n e v ω (4) whee + = s s n n E. (4) Note that in tems of and z, E is given by (7) as z E + =. (43)

9 Plane Stagnation Flows Conside a steady plane stagnation flow shown in Figue. F steady plane flow the Navie-Stokes equation educes to ( ψ, ψ) = ν ψ 4 (x, y) () ψ ψ ψ ψ = ν y x x y 4 ψ () y x Figue. Schematics of plane stagnation flow. Potential stagnation plane flow is descibed by U = ax, V = ay, ψ = axy. (3) We look f a solution of the fm Then ( y) ψ = xf. (4) u = xf, v = f, ψ = xf (5) Using (4) and (5), equation () becomes 9

10 ( 4 ) x xf f ff x = νf (6) ( 4) f f ff = νf. (7) Integating (7) we find That is f ff = νf + c (8) The bounday conditions ae: At y = 0, u = 0, v = 0 (9) At lage y, Equations in (3) holds. (0) ( 0) f ( 0) 0 f = =, () As y f ay () Using () we find Thus c = a (3) ff = νf a (4) f + Intoducing a change of vaiable (Schlichting,960) a η = y, f = aνϕ( η), (5) ν Equation (4) may be estated as ϕ + ϕϕ ϕ + = 0 (6) subject to bounday conditions ( 0 ) = φ ( 0) = 0 φ, (7) 0

11 as η ϕ = (8) Gaphical epesentation of the numeical solution is shown in Figue. Additional details of the solution ae discussed by Schlichting (960). Accdingly, At η =. 4, = u U ϕ, = φ ( η) (9) Hence, the bounday laye thickness is given by ν δ =.4 (0) a.6. Φ" Φ 0.8 Φ'=u/U η = a ν y Figue. Schematics of plane stagnation flow solutions.

12 Axisymmetic Stagnation Flow Potential axisymmetic stagnation flow is descibed as V = a, V z = az, ψ = a z. () The steady state Navie-Stokes equation f axisymmetic flows is given as whee ψ z ψ ψ 4 ( E ψ) + ( E ψ) + E ψ = νe ψ z z, () v z ψ =, v ψ =, (3) z E = +. (4) z Simila to the plane flow case we look f a solution of the fm with () z ψ = f, (5) v z = f, v = f, E ψ = f. (6) Using (5) and (6), Equation () educes to f ( 4) ( f ) f ( f ) f ( f ) = ν f (7) ff + νf"" = 0 (8) The bounday conditions ae v = v 0 at z = 0 ; ψ = a z as z (9) z = ( 0) = f ( 0) 0, f ( ) = a f = ψ = a z as z (30)

13 Integating (8) we find c (3) f ff = νf + F equation (3) to be valid at lage z, Intoducing a change of vaiables c = a a ξ = z, f = aνφ( ξ) (3) ν Equation (3) becomes subject to φ + φφ φ + = 0 (33) φ ( 0 ) = φ ( 0) = 0, ( ) = φ (34) The numeical solutions of the flow filed ae vey simila to the plane stagnation flow case shown in Figue with a slightly fulle velocity pofile. The details of the numeical solution ae discussed by Schlichting (960). 3