Lecture 7

Chapter 9. Incompressible Flow

Introduction:

In previous chapters, flows have been governed by mass and momentum equations only, density changes due to the flow have been insignificant, and the Mach numbers have been less than about 0.3. For compressible flows, energy and equation of state must be used to account for large changes in pressure, density, energy and entropy that occur for high Mach number flows. Review the derivations of the mechanical, general, and internal energy equations in Chapter 3 and in the math notes.

The first law of thermodynamics is applied to a system control volume, so that the heat conduction in is Q, the work out is W, the total energy change is E = Q - W (the first law of thermodynamics). Use an integral form for E = int [system CV] \rho e dV with e = u + ke + pe the total energy per unit mass, ke = v^2/2 is the kinetic energy per unit mass, and pe = gz is the potential energy per unit mass. Differentiate with respect to time using Leibnitz' rule and convert the surface integral to a volume integral of div [\rho e v], where \rho e v is the convective energy flux vector. Express the time derivative of Q as the negative of the conductive heat flux vector q = - k grad T dotted into the surface area vector dA of the control volume and integrated. From the divergence theorem dQ/dt = int [div k grad T] dV. The time derivative of the work out is dW/dt = dW_p/dt + dW_visc/dt . Move dW_p/dt in integral form dW_p/dt = int [system CV] \rho (p/\rho) v dV to the right hand side of the equation, convert to a volume integral and add to obtain int [div (u + ke + pe + p/\rho] dV. Recognize the enthalpy h = u + p/\rho to give int [div (h + ke + pe] dV. This form of the energy equation leads to the a new version of the Bernoulli equation: B'_1 = B'_2 + w_v - q, where B' = h + pe + ke, w_v = dW_v/dt / dm/dt, q = dQ/dt / dm/dt, and dm/dt is the mass flow rate at steady state through a stream tube with entrance 1 and exit 2.

Compressible flow regimes are:

• Ma < 0.3; incompressible flow
• 0.3 < Ma < 0.8; subsonic flow, no shock waves
• 0.8 < Ma < 1.2; transonic flow, shock waves
• 1.2 < Ma < 3.0; supersonic flow, no subsonic regions
• 3 < Ma; hypersonic flow

Gas properties:

Some gas properties are of importance, particularly the gas constant R and the specific heat ration k. For an ideal gas k = c_p / c_v = constant (1.4 for air). p = \rho RT, R = c_p - c_v , R = \Lambda / M_gas , \Lambda = 8314 m^2 / s^2 K .

For air:

• M = 28.97
• R = 287 m^2 s^-2 K^-1
• c_v = R/(k-1 )= 718 m^2 s^-2 K^-1
• c_p = kR/(k-1 ) = 1005 m^2 s^-2 K^-1

Isentropic processes:

From the first and second law of thermodynamics:

Tds = dh -dp/\rho

So:

s_2 - s_1 = c_p ln T_2 /T_1 - R ln p_2 / p_1 = c_v ln T_2 / T_1 - R ln \rho_2 / \rho_1

upon substitution of R = c_p - c_v

Giving:

p_2 / p_1 = (T_2 /T_1 )^k/(k-1) = (\rho_2 / \rho_1)^k

The speed of sound:

Consider one dimensional flow across a pressure wave;

The mass flow in is \rho A C and out is (\rho + \delta \rho) A (C - \delta V)

Combine this with the momentum balance;

pA - (p + \delta p) A = (\rho A C) (C - \delta V - C)

to give

C^2 = (\delta p / \delta \rho) (1 + \delta \rho / \rho )

where the sound speed a = (\delta p / \delta \rho) at constant entropy.

Adiabatic and isentropic steady flow:

The energy equation for steady state, steady flow in a stream tube (similar to Bernoulli's equation) is:

h_1 + 1/2 V_1^2 + g z_1 = h_2 + 1/2 V_2^2 + g z_2 -q + w_v

where q is the heat added per unit time to the stream tube by conduction divided by the mass flow rate through the tube, and w_v is the viscous working rate on the environment of the tube divided by the mass flow rate through the tube. For adiabatic flow, q = 0. For inviscid flow w_v = 0. Note that entalpy h = u + p/\rho, where u is the internal energy per unit mass. As an exercise, derive this energy equation from the first law of thermodynamics E = Q - W for a system.

Stagnation temperature:

An important reference condition in compressible flow is where the velocity is zero, giving "stagnation" temperature, enthalpy, etc., indicated by "0" subscripts. Another reference condition is where the velocity is sonic, indicated by * superscripts.

The normalized stagnation temperature T_0 / T is 1 + V^2 / 2c_p T from the energy equation.

Also:

T_0 / T = 1 + [(k-1)/2]Ma^2

because c_p T = [kR/(k-1)]T = a^2 /(k-1), where a^2 = kRT from the definition of the sound velocity a and the isentropic relationship between pressure and density for a perfect gas given above.

Isentropic pressure and density relations:

Since p_2 / p_1 = (T_2 /T_1 )^{k/(k-1)} it follows that:

*** p_2 / p_1 = [1 + [(k-1)/2]Ma^2]^{k/(k-1)}.

Since (T_2 /T_1 )^1/(k-1) = (\rho_2 / \rho_1) it follows that:

*** \rho_2 / \rho_1 = [1 + [(k-1)/2]Ma^2]^{1/(k-1)}