Stream Function, Velocity Potential, Flownet Construction

LECTURE #6 1.060 ENGINEERING MECHANICS II Conservation of Volume for a Streamtube (2-D Plane Flow) Q₁ (ms⁻¹) Q₂ (ms⁻¹) F E D C B A Streamtube coordinates: s is local direction of streamline and h is (local direction ⊥ streamline) ψ(ξ) = (ψs, ψh) = (ψs, 0) = ψs STREAM FUNCTION Define a function ψ such that [\frac{∂ψ}{∂s} = 0] [\frac{∂ψ}{∂h} = q_{s}] and therefore ψ = ψ(s, h) = ψ(h) = constant along s ψ - stream function is constant along a streamline. Velocity Potential Define a function ϕ such that ∂ϕ/∂s = -gs, ∂ϕ/∂h = 0 and therefore ϕ = ϕ(s,h) = ϕ(s) - constant along lines s ϕ = velocity potential is constant along lines s (streamlines (equipotential lines)) ∂ψ/∂h = -gs ψ2 - ψ1 = -gsh - Q12 = discharge in tube 1-2 ∂ϕ/∂s = -gs ∂ψ/∂h = -gs ψ2 - ψ1 = -gsh - Q12 = discharge in tube 1-2 Flow NET Lines of constant ψ (streamlines) and constant ϕ (equipotential lines) form two families of curves (s-lines and h-lines) that intersect each other at right angles, i.e. they form a rectangular pattern. If the two families of curves are constructed (drawn) such that they form a pattern of squares, i.e. Δs = Δh, the discharge in stream tubes formed by adjacent streamlines, Q12, is constant (as it should be). This is a FLOW NET. If we know the discharge, Q12, in a stream tube, we can obtain an estimate of the velocity from qs = Q12 / Δh where Δh (≈ Δs) is the side length of the square. Simple "Rules" to follow when Constructing Flow Nets 1) Use a pencil and bring an eraser. 2) Solid boundaries are streamlines. 3) Regions where flow is uniform - straight parallel streamlines - is a good place to start and end. 4) Sketch the streamlines and equipotential lines connecting regions identified in (3), observing the following: - ψ-lines form a square pattern. - Streamlines can never cross. - Streamlines can start and end only at flow boundaries. - Streamlines in the interior are guided by boundaries, but smoother (no kinks). 5) Examine sketch and repair where needed (this is where the eraser comes in!). 6) Discharge Q is constant within a tube. V = Q/(ΔsΔh) can be estimated at center of squares. Directed along ψ. 7) Pressure is estimated from Bernoulli along a streamline. STREAM FUNCTION & VELOCITY POTENTIAL Their Generalized Definitions and Limitations Differential form of Continuity Equation Volume Conservation: Rate in - Rate out = Rate of change Rate in - Rate out = 0 if incompressible fluid. ∂u/∂x + ∂v/∂y = 0 in 2-D ∂u/∂x + ∂v/∂y + ∂w/∂z = 0 in 3-D Stream Function ψ ψ = ψ(x, y, t) defined by u = ∂ψ/∂y; v = -∂ψ/∂x Continuity equation (in 2-D) is automatically satisfied since ∂u/∂x + ∂v/∂y = ∂²ψ/∂x∂y + ∂²ψ/∂y∂x = ∂²ψ/∂x∂y - ∂²ψ/∂x∂y = 0 Limitation of Stream Function Two-Dimensional Flows of an Incompressible Fluid