The Prandtl–Glauert transformation is a mathematical technique which allows solving certain compressible flow problems by incompressible-flow calculation methods. It also allows applying incompressible-flow data to compressible-flow cases.
Mathematical formulation
Inviscid compressible flow over slender bodies is governed by linearized compressible small-disturbance potential equation:
ϕ x x + ϕ y y + ϕ z z = M ∞ 2 ϕ x x (in flow field) {\displaystyle \phi _{xx}+\phi _{yy}+\phi _{zz}=M_{\infty }^{2}\phi _{xx}\quad {\mbox{(in flow field)}}}
together with the small-disturbance flow-tangency boundary condition.
V ∞ n x + ϕ y n y + ϕ z n z = 0 (on body surface) {\displaystyle V_{\infty }n_{x}+\phi _{y}n_{y}+\phi _{z}n_{z}=0\quad {\mbox{(on body surface)}}}
M ∞ {\displaystyle M_{\infty }} is the freestream Mach number, and n x , n y , n z {\displaystyle n_{x},n_{y},n_{z}} are the surface-normal vector components. The unknown variable is the perturbation potential ϕ ( x , y , z ) {\displaystyle \phi (x,y,z)} , and the total velocity is given by its gradient plus the freestream velocity V ∞ {\displaystyle V_{\infty }} which is assumed here to be along x {\displaystyle x} .
V → = ∇ ϕ + V ∞ x ^ = ( V ∞ + ϕ x ) x ^ + ϕ y y ^ + ϕ z z ^ {\displaystyle {\vec {V}}=\nabla \phi +V_{\infty }{\hat {x}}=(V_{\infty }+\phi _{x}){\hat {x}}+\phi _{y}{\hat {y}}+\phi _{z}{\hat {z}}}
The above formulation is valid only if the small-disturbance approximation applies,
| ∇ ϕ | ≪ V ∞ {\displaystyle |\nabla \phi |\ll V_{\infty }}
and in addition that there is no transonic flow, approximately stated by the requirement that the local Mach number not exceed unity.
[ 1 + ( γ + 1 ) ϕ x V ∞ ] M ∞ 2 < 1 {\displaystyle \left[1+(\gamma +1){\frac {\phi _{x}}{V_{\infty }}}\right]M_{\infty }^{2}<1}
The Prandtl–Glauert (PG) transformation uses the Prandtl–Glauert factor β ≡ 1 − M ∞ 2 {\displaystyle \beta \equiv {\sqrt {1-M_{\infty }^{2}}}} . It consists of scaling down all y and z dimensions and angle of attack by the factor of β , {\displaystyle \beta ,} the potential by β 2 , {\displaystyle \beta ^{2},} and the x component of the normal vectors by β {\displaystyle \beta } :
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