Chater 5 Simlifications to Conservation Equations 5.1 Steady Flow If fluid roerties at a oint in a field do not change with time, then they are a function of sace only. They are reresented by: ϕ = ϕq 1, q, q 3 Therefore for a steady flow ϕ t = 0. 5. One-, Two-, and Three-Dimensional Flows A flow is classified as one-, two-, or three-dimensional deending on the number of sace coordinates required to secify all the fluid roerties and the number of comonents of the velocity vector. For examle a steady three-dimensional flow requires three sace coordinates to secify the roerty and the velocity vector is given by: V = v 1 ê 1 + v ê + v 3 ê 3. Most real flows are three-dimensional in nature. On the other hand any roerty of a two-dimensional flow field requires only two sace coordinates to describe it and its velocity has only two comonents along the two sace coordinates that describe the field. The third comonent of velocity is identically zero everywhere. Steady channel flow between two arallel lates is a erfect examle of two-dimensional flow if the viscous effects on the lates are neglected. The roerties of the flow can be uniquely reresented by ϕ = ϕq 1, q and the velocity vector can be written as V = v 1 ê 1 + v ê. In addition ϕ q 3 = 0. The comlexity of analysis increases considerably with the number of dimensions of the flow field. In one-dimensional flow roerties vary only as a function of one satial coordinate and the velocity comonent in the other two directions are identically zero. All derivatives in the other directions are identically zero. In other words ϕ = ϕq 1, V = v 1 ê 1 and ϕ q = ϕ q 3 = 0. 5.3 Axisymmetric Flow In axisymmetric flow the variation of flow variables are zero in the direction of rotation but the velocity comonent in the rotation direction is not zero. For examle if the flow is symmetric about the q 1 axis and the lane containing the axis q 1 and q 3 are rotated in the direction of q then ϕ q = 0 but v 0. 5.4 Ideal Fluid Non-heat conducting, inviscid, incomressible, homogeneous fluid is defined as ideal fluid. The deendent variables of ideal fluid are and V. The equations of the Fluid flow are: 1 V = 0 V V t + V V = + f ρ 9
If we consider conservative body forces only f = U, then the above equation becomes: V V t + V V = + U ρ Rearrange the above equation as: t + ρ + V V U V V = 0 The above equation is valid at any oint in an ideal fluid and can be integrated in closed form for two situations. 1. Steady flow along a streamline.. Unsteady irrotational flow. 5.5 Streamlines and Stream Function 5.5.1 Streamlines A streamline is defined as an imaginary line drawn in the fluid whose tangent at any oint is in the direction of the velocity vector at that oint. By definition there is no flow across it at any oint. Any streamline may be relaced by a solid boundary without modifying the flow. Any solid boundary is itself a streamline of the flow around it. 5.5. Pathline This is the ath traced out by any one article of the fluid in motion. In unsteady flow, the two are in general different, while in steady flow both are identical. 5.5.3 Equation for A Streamline ds V = 0 V = V 1 ê 1 + V ê + V 3 ê 3 ds = h 1 dq 1 ê 1 + h dq ê + h 3 dq 3 ê 3 ds V ê 1 ê ê 3 = h 1 dq 1 h dq h 3 dq 3 V 1 V V 3 = 0 V 3 h dq V h 3 dq }{{} 3 ê 1 + V 1 h 3 dq 3 V 3 h 1 dq 1 ê }{{} + V h 1 dq 1 V 1 h dq ê }{{} 3 = 0 =0 =0 =0 Differential equations V 3 h dq V h 3 dq 3 = 0 V 1 h 3 dq 3 V 3 h 1 dq 1 = 0 V h 1 dq 1 V 1 h dq = 0 Symmetric form h 1 dq 1 = h dq V 1 V }{{ } D = h 3dq 3 V 3 30
From the symmetric form in -D: where ds ds 1 is the sloe for the line. Also if V = V 1 ê 1 + V ê then h dq h 1 dq 1 = V V 1 = ds ds 1 V V 1 = tan θ which is the angle of the velocity vector The equation of the streamline ds V = 0 imlies that the sloe of the streamline is equal to the angle of the velocity vector at that oint. Hence, the velocity vector at any oint on the streamline is a tangent to the streamline. 5.5.4 Stream Function From the symmetric form in -D: Integration yields: h dq h 1 dq 1 = V V 1 q = fq 1 or F q 1, q = C because V 1 = V 1 q 1, q and V = V q 1, q. Let us say that F is called a stream function ψ, or ψ = ψq 1, q = C - a stream function for comressible flows. Different constants of integration yield different streamlines. Figure 4.1: Stream lines Let ab, cd reresent two streamlines. No fluid asses ab or cd. Therefore the same mass of fluid must cross gh and ef. If the streamline ab is arbitrarily chosen as a base, every other streamline in the field can be identified by assigning to it a number ψ equal to the mass of fluid assing, er second er unit deth erendicular to the lane containing the base streamline and the streamline in question. ψ = C C 1 = ψ ψ 1 = e f ρ V ê n dl = ρv n l where V n is the normal comonent of velocity and l is the normal distance between streamlines. or ψ = ρv n l or ψ = ρv n l and in the limit l 0 ψ = ψ = ρv n l l Thus the velocity comonent in any direction is obtained by differentiating ψ at right angles to that direction. 31
This stream function is defined for two-dimensional flow only. In general, it is not ossible to define a stream function for three dimensional flow, though there is a secial form, for axi-symmetric flows known as the Stokes stream function. 5.6 Relation Between ψ and V 5.6.1 Derivation from The Physical Meaning Conventions: Direction of integration for the chosen coordinate system is ACW. Do all derivations in the first quadrant with x, y and all velocity comonents u, v or v r, v θ being ositive. The sign convention yields ositive for flow going out and negative for flow going in. In line integrals the integral is ositive if the flow is left to right if you look in the direction of integration. 5.6. Cartesian Coordinate System Figure 4.: Velocity comonents between stream lines Mass flow across ef: Comaring equation [1] and [] we get: ef = ψ = e 1 e f ρv dx + or ψ = ρv x + ρu y e 1 ρu dy lim d ψ = ρv dx + ρu dy [1] ψ 0 Since ψ = ψx, y d ψ = ψ ψ dx + dy [] x ρu = ψ ψ ; ρv = x comressible flow For incomressible flow: u = ψ/ρ v = ψ/ρ x = ψ = ψ x 3
5.7 Stream Function 5.7.1 Ex Given: -D incomressible flow { u = x v = 6x y dx = dy u v dx x = dy not a variable searable 6x y ψ = u = x, ψ = xy + fx + C 1 ψ = v = 6x + y = y + f x + 0 x f x = 6x, fx = 3x + C ψ = xy + 3x + C 5.8 Vorticity, Circulation & Stokes Theorem 5.8.1 Vorticity Vorticity is defined as twice the angular velocity. ξ = w = V In 3-D Cartesian coordinates w = w x î + w y ĵ + w zˆk { w v z w = 1 î + u z w v ĵ + x x u } ˆk 5.8. Irrotational Flow The flow is defined irrotational if V = 0. 1. V = 0 at every oint in the flow then the flow is irrotational.. V 0 at any oint the flow is rotational. General Curl A = A 1 h 1 ê 1 h ê h 3 ê 3 = h 1 h h 3 q 1 q q 3 h 1 A 1 h A h 3 A 3 5.8.3 Circulation Circulation is defined as the line integral of the velocity around any closed curve. Γ = V d l C Circulation is a kinematic roerty that deends only on the velocity field and the choice of the curve C. When circulation exists in a flow it simly means that the line integral Γ = V d l is finite. C 33
5.8.4 Stokes Theorem The line integral of a vector V over C is equal to the surface integral of the normal comonent of the curl of V over S. V d l = O V ds C or Γ = S V d l = O V ds S 1. φ exists if and only if V d l = 0 C. If C V d l = 0, it does not imly φ exists. V = φ if V = 0 5.9 Bernoulli s Equation for A Steady Flow Along A Streamline For a conservative body force field the equation of motion for an ideal fluid flow is: t + ρ + V V U V V = 0 For a steady flow the above equation becomes: ρ + V V U V V = 0 If we scalar multily both sides of the above equation by ds we get: ρ + V V U ds V V ds = 0 ds Using the definition of the streamline ds V = 0 the second term on the left hand side of the above equation goes to zero reducing to: ρ + V V U ds = 0 From the definition of directional derivative the above equation becomes: d ρ + V V U = 0 which uon integration yields the Bernoulli s equation along a streamline: ρ + V U = constant If the body force f is 0, 0, g then U = gz in Cartesian coordinates and the Bernoulli equation becomes: ρ + V + gz = constant 34
5.10 Bernoulli s Equation for Irrotational Flow For irrotational flow V = 0 equation of motion becomes: V V t + V V }{{ = + f } ρ =0 For steady flow t = 0, the above equation becomes: V = + f ρ If we consider conservative body forces only f = U, then: ρ + V U = 0 Take a dot roduct with d l, an elemental length along any arbitrary ath: [ ρ + V ] U = 0 d l For gravitational body force U = gz: 5.11 Potential Flow d l = d [ d ρ + V ] U = 0 ρ + V U = constant ρ + V + gz = constant Bernoulli s eqn. valid for ideal, irrotational, steady flow Non-heat conducting, inviscid, incomressible, and irrotational flow of a homogeneous fluid is defined as otential flow. The deendent variables of ideal fluid are and V. The equations of the Fluid flow are: 1 V = 0 V V t + = + f ρ If we consider conservative body forces only f = U, then the above equation becomes: V V t + = + U ρ Rearrange the above equation as: t + ρ + V V U = 0 35
5.1 Velocity Potential φ Velocity otential is defined only for ideal irrotational flow for steady or unsteady flow as: V = φ V = 1 h 1 φ q 1 ê 1 + 1 h φ q ê = v 1 ê 1 + v ê v 1 = 1 h 1 φ q 1 and v = 1 h φ q φ is defined for -D or 3-D and for unsteady flow. ψ, stream function is defined only for steady -D or axisymetric flows as long as the flow is hysically ossible. 5.13 Lalace Equation Irrotational and incomressible flow. From the mass conservation equation ρ t + ρ V = 0 Since ρ is constant ρ t = 0 ρ V = ρ V = 0 If the flow is irrotational V = φ. or V = 0 V = φ = 0 = φ Lalace Equation 5.13.1 Cartesian φ = φ x + φ + φ z 5.13. Cylindrical φ φ = = 1 r r r êr + φ r θ êθ + φ r φ r z êz + 1 r φ θ + φ z = 0 [ êr ê θ ] [ cos θ sin θ = sin θ cos θ ] [ î ĵ ] ê r θ ê θ θ = ê θ = ê r 36
5.13.3 Irrotational -D v x u = 0 ψ x x ψ x + ψ = ψ = 0 ψ = 0 Lalace equation has solutions which are called as harmonic functions. For -D flow 1. Any irrotational and incomressible flow has a velocity otential φ and stream function ψ that both satisfy Lalace equation.. Conversely any solution reresents the velocity otential φ or stream function ψ for an irrotational and incomressible flow. A owerful rocedure for solving irrotational flow roblems is to reresent φ and ψ by linear combinations of known solutions of Lalace equation. φ = C i φ i, ψ = C i ψ i Finding the coefficients C i so that the boundary conditions are satisfied both far from the body and the body surface. Say φ 1 and φ are solutions of φ = 0, therefore φ 1 = 0; φ = 0 A 1 φ 1 = 0 or A 1 φ 1 = 0 Similarly A φ = 0 Therefore A 1 φ 1 + A φ = 0 φ = A 1 φ 1 + A φ is also a solution. A comlicated flow attern for an irrotational and incomressible flow can be synthesized by adding together a number of elementary flows which are also irrotational and incomressible. 5.14 Boundary Conditions 5.14.1 Infinity Boundary Conditions V 8 V sinα 8 α V cosα 8 V = V cos αî + V sin αĵ 37
u = φ x = V cos α = ψ v = φ = V sin α = ψ x The coordinate axes are attached to the body. 5.14. Wall Boundary Conditions At the body, the velocity must be tangential to the surface, that is, a streamline must conform to the contour of the body. ψ surface = constant ψ or s = 0 where s is the distance measured along the body surface. 5.14.3 Streamline V ds = 0 u dy v dx = 0 dy v = dx u surface surface 5.14.4 Solid Body Comonent of velocity normal to the surface is zero. V ˆn = 0 φ ˆn = 0 φ or n surface = 0 38