Exercise 4: Adimensional form and Rankine vortex. Example 1: adimensional form of governing equations

Size: px

Start display at page:

Download "Exercise 4: Adimensional form and Rankine vortex. Example 1: adimensional form of governing equations"

Stewart Lee
5 years ago
Views:

1 Fluid Mechanics, SG4, HT9 Septembe, 9 Execise 4: Adimensional fom and Rankine votex Example : adimensional fom of govening equations Calculating the two-dimensional flow aound a cylinde (adius a, located at x = y = ) in a unifom steam U involves solving ū t + (ū )ū = ρ p + ν ū, ū =, with the bounday conditions ū = on x + y = a ū (U,) as x + y. Rewite this poblem in non-dimensional fom using the dimensionless vaiables x = x/a, ū = ū/u, p = p/ρ, t = tu/a. Note that the scaling x = x/a implies = a and t = tu/a gives Change to dimensionless vaiables and divide by /a a The Reynolds numbe is Re = The continuity condition is simply t = a U t. ū t + U a (ū )ū = ρu ρa p + νu a ū ū t + (ū )ū = p + a inetial foces = ν U viscous foces = U a ν a ν U a a Re ū ū t + (ū )ū = p + Re ū ū = U a ū = ū =. The bounday conditions ae non-dimensionalised as follows: ū = on x + y = a Uū = on a x + a y = a ū = on x + y =. ū (U,,) as x + y Uū (U,,) as a x + a y ū (,,) as x + y A solution to this poblem will depend on x, Re and t only. The solution is thus the same fo a specific Re independently of the individual values of U, a and ν.

2 Example : voticity and defomation Show that the net viscous foce pe unit volume is popotional to the spatial deivative of voticity, i.e. τ ij x j = µε ijk ω k x j and discuss its implication fo flows with unifom voticity (as in solid-body otation). τ ij = µ ( ui + u ) ( j ) u i = µ + u j = µ u i x j x j x j x i x j x j x j x i x j x j ( ) ω k u m u m µε ijk = µε ijk ε klm = µε kij ε klm = µ(δ il δ jm δ im δ jl ) u m = x j x j x l x j x l x j x l ( ) u j µ u i = µ u i x j x i x j x j x j x j Thus τ ij x j = µε ijk ω k x j The net viscous foce vanishes when the voticity is unifom, since no defomation exists. The Rankine votex A simple model fo a votex is given by the combination of a igid-body otation within a coe, and a decay of angula velocity outside. This can be descibed by ω, < a, u θ = ωa, > a, u = u z = and is called a Rankine votex. 3 u θ ω 3 3 Figue : Velocity and voticity in a Rankine votex with ω = a =. Example 3: Rankine votex Conside the Rankine votex descibed above. a) Find the pessue inside and outside of a Rankine votex

3 We use the Eule equations fo incompessible flow, i.e. neglecting viscous effects. Dū Eule equations Dt = ρ p + ḡ ū = Dū Dt = ū +(ū )ū t = We ae woking pefeably in cylindical coodinates, use the fomulas given in the lectue notes: (ū )ū = u θ Inset this into the Eule equations: u θ u θ θ = ū = u + u θ θ (u θē θ ) = u θ u θ θ ēθ + u θ θ + u z z p = p + p + p ē θēθ zēz ē θ u θ ē = ρ ( p ē + ē θ u θ = u θ u θ θ θ ēθ u θ ē = ē p + p ) gē z θēθ zēz Look at the diffeent components: ē θ : Solve fo the pessue when < a: ē : u θ = p ρ = p ρ θ p = p(,z) ē z : = p ρ z g only. ē : ω = p ρ p = ρω + f(z) ē z : p z = ρg f(z) = ρgz + C So we obtain fo the pessue: Solve fo the pessue when > a: p(,z) = ρω ρgz + C fo < a ē : ω a 4 3 = p ρ p = ρω a 4 + f(z) ē z : p z = ρg f(z) = ρgz + C 3

4 So we obtain fo the pessue: p(,z) = ρω a 4 ρgz + C fo > a Now detemine the diffeence between the constants C and C by evaluation at = a: ρω a ρgz + C = ρω a ρgz + C C C = ρω a b) Detemine the pessue diffeence p between = and p = p p = ρgz + C ( ρgz + C ) = C C = ρω a 4

5 c) Calculate the shape of a fee suface at atmospheic pessue p. Find the diffeence in z between = and { = : p = ρgz + C z z = C C = ω a : p = ρgz + C ρg g Detemine the shape of the fee suface: { p = ρω ρgz + C < a z z = ω g + C p ρg p = ρω a 4 ρgz + C > a z z = ω a 4 g + C p ρg Set z = at =. Then C = p and we futhe get z = ω g < a and z = ω a 4 g + C C = ω a 4 ρg g + ω a = ω a ) ( a g g > a So we have z() = ω g ω a g < a ( ) a > a. z Figue : The fee suface of a Rankine votex with ω = a = and g =

Stress, Cauchy s equation and the Navier-Stokes equations

Stress, Cauchy s equation and the Navier-Stokes equations Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted