Navier-Stokes Equations: Potential Flows - Prandtl-Glauert Similarity Laws

Sketch of the pressure coefficient vs the freestream Mach number

P-G Singularity

Navier-Stokes Equations
Potential Flows: Prandtl-Glauert Similarity Laws

Here I'll give a brief discussion of the similarity laws for linearized subsonic and supersonic flows. To keep the discussion as simple as possible, I'll restrict attention to pure two-dimensional flows. Thus, a reasonable starting point is the Thin Airfoil Equation.

The standard way to derive these similarity relations is to non-dimensionalize the equations (Pg6)-(Pg9). The resultant non-dimensional equations will be recognized as well known canonical equations. Furthermore, the non-dimensionalization will reveal how the flow details will depend on the physical parameters of the flow, e.g., the Mach number, the thickness or the angle of attack of the wing, and the freestream flow speed. The latter feature is likely to be more important in the design process than the numerical details of the flow.

We begin by adding the rather mild constraint that the shape of the wing given in our Introduction is of the form f = tF(x/L). Here t is a measure of the thickness of the wing having units of length, F is a non-dimensional function of the non-dimensional variable x/L and L is a measure of the length or chord of the wing in the flow direction. As pointed out above, we are taking the flow to be two-dimensional; thus, we have dropped any dependence on the z direction.

We now change variables from the dimensional variables x, y, j to the non-dimensional variables

x º x/L,

(Ps1)

h º | 1-(M_¥)² |^½ y/L,

(Ps2)

F º | 1-(M_¥)² |^½ j/Ut,

(Ps3)

where F is the scaled, non-dimensional velocity potential and should not be confused with the viscous dissipation found in our discussion of the full Navier-Stokes equations. The scalings (Ps1)-(Ps3) are known as the Prandtl-Glauert scalings.

When (Ps1)-(Ps3) are substituted in (Pg6)-(Pg8) we find that

sgn( 1 - (M_¥)² ) F_xx + F_hh » 0,

(Ps4)

F_h » F_x on h » 0,

(Ps5)

F_x, F_h ® 0 as x ® - ¥,

(Ps6)

where sgn( 1 - (M_¥)² ) is just the sign of the difference 1 - (M_¥)² and is 1 if the freestream flow is subsonic, i.e., M_¥ < 1, and is -1 if the freestream flow is supersonic, i.e., M_¥ > 1.

The pressure coefficient for thin airfoils, i.e., (Pg9), can be written:

C_p

        -2 e F_x

(Ps7)

=   ----------------- ,

      | 1 - (M_¥)² |^½

which holds for all x and h. In (Ps7) e º t/L « 1 is a non-dimensional parameter measuring the slenderness of the airfoil.

If we choose either a subsonic ( sgn( 1 - (M_¥)² ) = 1 ) or supersonic ( sgn( 1 - (M_¥)² ) = -1 ) flow, the solution to (Ps4)-(Ps6) for F depends only on the generic shape of the wing through F(x) and not on the thickness or the Mach number. Thus, (Ps7) provides us with the explicit dependence of C_p, and therefore the pressure, on the principal physical parameters of the problem without having to solve the boundary value problem (Ps4)-(Ps6). Because F_x is generally non-zero, we may also conclude that the pressure perturbations ® ¥ as M_¥ ® 1. If we examine equations (Pg10)-(Pg11) we conclude that the density and temperature perturbations also become unbounded as M_¥ ® 1. The singularity caused by the |1 - (M_¥)²|^½ term in the denominator of (Ps7) is referred to as the Prandtl-Glauert singularity. A discussion of its role in the formation of transonic condensation can be found here.

The non-dimensional equations (Ps4)-(Ps7) are the core of the Prandtl-Glauert similarity rules. In the remainder of this section I'll describe the details of the two special cases of subsonic and supersonic flow.

Subsonic Flow
In the case of subsonic flow M_¥ < 1 and sgn( 1 - (M_¥)² ) in (Ps4) is equal to 1. Equation (Ps4) then reduces to Laplace's equation, solutions of which are well known to many undergraduates in math, physics, and engineering. Because (Ps4)-(Ps6) are now completely independent of M_¥, all subsonic flows for a given wing shape, i.e., a given F(x), are determined by solving (numerically, analytically, or experimentally) the boundary value problem (Ps4)-(Ps6) once. Solutions for various subsonic Mach numbers are then obtained by applying the scalings (Ps1)-(Ps3). Another way of stating this result is to say that all subsonic flows are just stretched incompressible (M_¥ = 0) flows.

When a collection of flows are simply stretched versions of each other we say that the flows are self-similar.

In subsonic flows we can also simplify the expression for the pressure coefficient (Ps7). If we recognize that F_x is independent of M_¥, it can be shown that

C_p

         C_pinc

(Ps8)

=   ----------------- ,

      ( 1 - (M_¥)² )^½

where C_pinc is the pressure coefficient corresponding to the same wing in an incompressible (M_¥ = 0) flow. Result (Ps8) is particularly useful for evaluating the pressure on the airfoil where h » 0. It can also be shown that the lift of a two-dimensional airfoil in subsonic flow also increases due to the Prandtl-Glauert singularity seen in (Ps8) or (Ps7).

Supersonic Flows
When the flow is supersonic M_¥ > 1 and sgn( 1 - (M_¥)² ) in (Ps4) is equal to -1. Equation (Ps4) then reduces to the linear wave equation, the solutions of which are also well known. A typical solution in the upper half plane (h > 0) for a single wing is just a simple-wave solution of the form:

F = - F(x - h)

and F is seen to be constant on the right-running Mach lines given by x - h = constant. We can also show that the streamlines, i.e., particle paths, are simply shifted versions of the wing shape.

As in the case of subsonic flow, all supersonic flows corresponding to a fixed F are identical up to the stretchings (Ps1)-(Ps3). A nice result is that the scalings automatically change the slope (or Mach angle) of the Mach waves without needing a separate derivation.

As in subsonic flow, the pressure, temperature and density perturbations are singular as M_¥ ® 1.

Conclusion
The above discussion demonstrates that subsonic and supersonic flows can be described by one of two well known canonical equations. In subsonic flow the governing equation is the (elliptic) Laplace equation and, in the case of supersonic flow, the equation is the (hyperbolic) wave equation. The difference in behavior of the fluid in these two regimes is simply due to the difference in the nature of elliptic and hyperbolic flows.

Flows at different Mach numbers within these two Mach number ranges are simply stretched versions of each other. Thus the result of our non-dimensionalization has been to demonstrate the self-similarity of the flow.

The above conclusions are based on a simple mathematical analysis of the thin airfoil problem. They are valid as long as the exact (full) potential equation can be linearized. Even when the wing is thin, the linearization breaks down in the case of transonic flow (M_¥ » 1) and in the case of hypersonic (M_¥ ® ¥) flow. The Prandtl-Glauert singularity seen in equations (Ps7) or (Ps8) clearly suggests that the linearization ought to break down in the transonic regime. A clue that the linearization breaks down at large Mach numbers can be seen by examining the angle of the Mach lines. It turns out that the Mach lines (the lines along which bits of the solution propagate) will disappear inside the wing at sufficiently large M_¥. It turns out that both transonic and hypersonic flows have their own, somewhat more complicated, similarity laws.

In closing, it should also be noted that other similarity laws can be derived for other aerodynamic flows. Examples include the similarity laws for axisymmetric bodies and for fully three-dimensional wings. In the latter case, we simply need to account for a length scale measuring the span. The new feature of the similarity will be a similarity parameter related to the wing aspect ratio. An excellent introductory treatment of various similarity laws can be found in Ashley and Landahl's Aerodynamics of Wings and Bodies. You can find the full reference to this text in the compressible flow section of my Great Books list.