Fluid Mechanics and Potential Flow

Potential flow fundamentals

Potential flow is an idealized model in which the fluid is inviscid and the flow is irrotational, so the velocity can be written as the gradient of a scalar velocity potential. Defining \mathbf{V} = \nabla\varphi gives u = \partial\varphi/\partial x and v = \partial\varphi/\partial y in two dimensions. For incompressible flow, continuity reduces to the Laplace equation \nabla^2\varphi = 0. Because this equation is linear, elementary solutions such as a uniform stream, a source or sink, a doublet and a vortex can be superposed to build more complex body shapes. To identify the elementary flow that a given potential represents, differentiate it and examine the velocity components. A potential that is linear in x and y gives constant velocity components, which is a uniform flow, since \varphi = Ax + By gives u = A and v = B everywhere. The doublet is the limit of a source and a sink of equal strength placed at a small separation, with the product of strength and separation held fixed as the separation goes to zero. Its potential is proportional to \cos\theta/r. Superposing a doublet on a uniform stream gives the classic flow past a circular cylinder, whose surface velocity is v_\theta = -2U_\infty\sin\theta. The surface speed therefore reaches a maximum of 2U_\infty at the top and bottom of the cylinder and vanishes at the front and rear stagnation points. A well-known limitation is d'Alembert's paradox. Inviscid potential theory predicts zero drag on a body in steady uniform flow, because the symmetric pressure distribution integrates to zero net force. Measured drag is finite, and the discrepancy is attributed to viscosity and the separation of the boundary layer from the body. Performance relations that often appear alongside these topics should be kept separate. The absolute ceiling, where the maximum climb rate is zero, is always higher than the service ceiling, where the climb rate has fallen to a small specified value. Stall speed in equivalent airspeed is constant, but the true airspeed at stall is V_{TAS} = V_{EAS}/\sqrt{\sigma}, so it rises with altitude as the density ratio \sigma falls. A tailwind adds to the ground distance covered, so range increases. For a jet flying at constant altitude, range depends on air density, so an altitude of maximum range exists for a given weight.

Dynamic similarity and drag scaling

Dynamic similarity means that a model and its prototype share the same values of all governing dimensionless groups, so their flow patterns and dimensionless force coefficients match. For viscous-dominated external flows such as drag on bodies, the controlling group is the Reynolds number Re = \rho V L/\mu = VL/\nu. When Re is matched, the drag coefficient C_D = D/(\frac{1}{2}\rho V^2 A) is the same for model and prototype. Unsteady phenomena such as vortex shedding also require the Strouhal number St = fd/V to match, so the shedding frequency scales as f \propto V/d. For a model tested in a different fluid or at a different scale, matching Re gives V_m = V_p (L_p/L_m)(\nu_m/\nu_p), so the kinematic viscosity ratio enters the speed scaling directly. Because drag scales as D \propto \rho V^2 L^2, the force ratio is F_p/F_m = (\rho_p/\rho_m)(V_p/V_m)^2(L_p/L_m)^2. In the same fluid, matching Re forces V \propto 1/L, so the drag ratio becomes unity and a smaller model must simply be tested at a higher speed. For wind-tunnel wing tests in the same tunnel at the same Re, the chord sets the speed through V_2 = V_1 c_1/c_2. For a rectangular wing of fixed span, the chord equals the span divided by the aspect ratio, c = b/AR, so a change in aspect ratio changes the chord and therefore the required speed. Buckingham's pi theorem sets the size of a dimensional test programme. With n variables involving m fundamental dimensions, the number of independent dimensionless groups is n - m, and a programme that varies each group over k values needs k^{n-m} runs. For an airfoil in steady incompressible flow, the variables are speed, density, viscosity, chord and angle of attack, so there are five variables and three dimensions. Two groups remain, Re and the dimensionless angle of attack, so ten values of each give 10^2 runs.

Momentum theorem for control volume

For a steady control volume, the momentum theorem states that the sum of external forces equals the net outflow of momentum through the control surface, \sum \mathbf F = \oint \rho\mathbf V(\mathbf V\cdot\mathbf n)\,dA. A common application is a jet. The mass flow is \dot m = \rho A V, and the force needed to hold a device against the jet equals \dot m times the change in the jet velocity component along the force direction. A jet deflected through an angle leaves with a reduced horizontal component, so the horizontal reaction is \dot m times the difference between the entering and leaving horizontal velocities. Atmospheric pressure acting uniformly on the whole control surface contributes no net force, which is why the pressure term can be dropped in such problems. When the velocity profile at a boundary is not uniform, mass conservation still requires inflow and outflow fluxes to balance, and the momentum flux must be integrated over the profile. For a power-law profile u = A_0(y/h)^n across a section of height h, the integral \int_0^h (y/h)^n\,dy = h/(n+1) gives the flux. If the flux through one boundary must vanish, the outflow through the other must equal the inflow, which fixes the amplitude A_0 in terms of the uniform speed. Momentum theory also gives drag from a wake survey. When the static pressure is uniform on all surfaces of a control volume enclosing a body, the drag per unit span equals D' = \rho\int u(U_\infty - u)\,dy across a downstream section, which is the momentum deficit of the wake. The drag coefficient is then C_d = D'/(\frac{1}{2}\rho U_\infty^2 c) for chord c.

Blasius laminar boundary layer

The Blasius solution describes the steady, incompressible, laminar boundary layer on a flat plate with zero pressure gradient. Its key results are \delta \approx 5x/\sqrt{Re_x}, \theta \approx 0.664\,x/\sqrt{Re_x} and \delta^* \approx 1.72\,x/\sqrt{Re_x}, with Re_x = Ux/\nu. Since \delta \propto \sqrt{\nu x/U}, the thickness grows as the square root of distance from the leading edge, and the momentum thickness follows the same x^{1/2} law. A higher free-stream speed gives a thinner laminar layer at a fixed station, so \delta \propto U^{-1/2}. Writing the result in terms of the local Reynolds number, the ratio \delta/x = 5/\sqrt{Re_x} is proportional to Re_x^{-1/2}. Growth exponents must not be confused with the turbulent case, where the thickness grows approximately as x^{4/5} and the velocity profile is commonly approximated by a one-seventh power law. Using a turbulent exponent or an inverse-square law for a laminar layer is a common error.