Lecture 26: Fluid Flow. Chapter #14: Fluid Flow. We will treat two types of Fluid flow problems that are governed by identical differential equations. Flow through a porous medium: solve for a fluid head. Idealized steady state fluid flow (inviscid, incompressible, and irrotational): solve for a velocity potential. In both cases we will use a linear approximation to the quantity of interest. The solution process is very similar to that used in the Prandtl Stress problem (yet to come). We will employ a weighted residuals scheme to set-up the discrete equations. We will solve for the head/potential at the node points. We will recover a physical quantity (flow velocity) from the <a href="/keyword/finite-element/" >finite element</a> solution. MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. 14.1. Derivation of the Basic Differential Equations. First, note the restrictions on the analysis. The fluid is incompressible, inviscid, and irrotational. The problems we consider do not include energy considerations. There is no viscous work and no heat transfer to the fluid. In MECH 420 we consider the fluid &quot;free stream&quot; and thus can't accurately capture viscous drag near the boundaries of objects. Viscous effects induce rotation at the boundaries of solid bodies (shear layers) which is the precursor to separation and the associated spike in pressure drag. We only consider flow fields that have achieved a steady state. The values of the discrete nodal heads and potentials are constant (the streamlines are not changing). MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. Consider the actual flow around an airfoil: y Ma 0.3 (incompressible). Control volume in the free-stream (the truly inviscid portion of the field problem) x z MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications In regions where shear (the no slip condition) induce rotation a correction needs to be superposed over our inviscid, irrotational approximation. Lecture 26: Fluid Flow. Assume the BC's ensure a 1-D fluid flow. Consider conservation of mass through the control volume. M in + M generated = M out vx Adt + Qdt = vx + dx Adt Q internal volumetric source (m3 /s). A cross sectional area of the duct/channel (m 2 ). What is vx(x)?... MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. There are two possibilities for the definition of the fluid velocity: 1) Flow in pipes/ducts and external flows: Irrotational flow: v = 0 v = - velocity potential For the 1-D (x dimension) problem: vx = - d dx Control volume of the analysis. MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. 2) Flow through a porous medium: apply an empirical relationship between the fluid head and velocity Darcy's law. Kxx is the permeability of the medium. It is an emperically defined value that relates pressure drop to the velocity achieved. For the 1-D (x dimension) problem: d vx = - K xx (dimensionless) dx Fluid head (pressure and gravity heads) (m). d = g x hydraulic gradient dx MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. Regardless of the problem type, we define the velocity field in a consistent form: vx = - K xx d dx For free stream problems the `permeability' is unity. At the right hand (downstream) boundary of the 1-D control volume: vx + dx = - K xx d d d - K xx dx dx dx dx Substituting in the form of the boundary velocities (mass fluxes): d d K xx ( Adxdt ) + Q ( dt ) = 0 dx dx MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. Dividing through by the constant factor Adxdt to isolate the second order differential term: d d K xx +Q = 0 dx dx Q normalized volume delivery (m3 of fluid per m3 of element per second: s -1 or Hz) Note that the form of the differential equation above is the same as that for 1-D heat conduction without convection. d dT K xx +Q = 0 dx dx MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. 14.2. One Dimensional <a href="/keyword/finite-element/" >finite element</a> Formulation. Our approximation to the head/potential is linear: ^ pj pi p ^ = N N i i j pj ^ x Ni = 1 - L ^ x Nj = L ^ x L MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. Using the same linear approximation to the fluid head/velocity potential function, (x), that we applied for temperature variation in Chapter #13: d - K xx dx x =0 1 K xx 1 -1 pi 1 ^ = QL + 1 L -1 1 p j 2 d + K xx dx x = L ^ Boundary conditions: = B vx x = vn B A value of the velocity potential at the boundary in question. The velocity of the fluid crossing the element boundary. MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. ( x) = N ^ x N = 1 - L pi = pj ^ x L d = vx x =0 = vni ^ dx x =0 ^ d + K xx = -vx x = L = vnj ^ dx x = L ^ - K xx K xx L 1 -1 i 1 1 vni = QL + -1 1 2 j 1 vnj i j vn = incoming velocity. Positive is entering the element and negative if flow is leaving the element. L ^ x=0 MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications ^ x=L Lecture 26: Fluid Flow. 14.3. Two Dimensional <a href="/keyword/finite-element/" >finite element</a> Formulation. For 2-D problems we will only consider free stream problems (external flows around bodies). Extending the 14.1 material by analogy to the material of 13.1 (with no convection) the 2-D governing equation is: + K yy K xx +Q = 0 x x y y 2 2 + 2 +Q = 0 2 x y Continuity equation for an incompressible and irrotational fluid in two dimensions. Permeability is unity. Can consider a planar <a href="/keyword/finite-element/" >finite element</a> to solve this 2-D problem. Use the triangular element applied in heat transfer. Now we are interpolating nodal values, pi, pj, and pm. MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. Again by analogy to the 2-D heat conduction problem (with no convective type terms)... y x ( x, y ) = N N = Ni pi = pj p m Nj N m ; Ni = 1 ( i + i x + i y ) 2S A linear approximation to a function that is differentiated twice in the governing equations. We integrate by parts (green's theorum) to form boundary terms on the normal velocities across the element edges... MECH 420: <a href="/keyword/finite-element/" >finite element</a> Applications Lecture 26: Fluid Flow. The final form of the 2-D Fluid flow element equation is: 1 4S i j m i K xx j 0 m pi 0 i j m p = K yy i j m j pm 1 v L 1 v L QS nij ij n jm jm 1 + 1 + 2 3 2 1 0 0 1 + vnmi Lmi 2 1 1 0 1 Note: as in 13.5 we have normalized against the element thickness. The thickness does not...