How can one estimate wave drag?

Wave drag is an especially high drag above mach 1, but I've found it hard to find source materials to estimate it. I'm looking to estimate how to design better for wave drag by estimating how it would affect my flight in the transonic region Mach 0.8 - 1.2.

• Welcome to Aviation.SE. Speeds up to Mach 1 are subsonic by definition and do not have wave drag. You might want to simplify your question to focus on one aspect. It requires several answers in its current form, and this site tends to like one at a time. Sep 18, 2018 at 18:15
• Hey @Pilothead, thanks for the heads up. Thanks for welcoming me , and I have simplified the question and perhaps I'll add the other elements in another question. Sep 19, 2018 at 1:28
• @Pilothead As you increase speeds past about Mach 0.6 you enter the transonic regime where areas over which flow acceleration occurs (such as wings) may encounter sonic or supersonic flow Dec 22, 2023 at 16:22

Usually wave drag $$c_{d_w}$$ starts becoming relevant at the drag divergence Mach number $$M_{DD}$$, which is defined as

$$\left.\frac{\partial c_{d_w}}{ \partial M}\right\vert_{M=M_{DD}}=0.1 \tag{1}$$

Or in words, the Mach number at which the wave drag coefficient versus M has a gradient of 0.1

Lock$$^1$$ has derived an empirical formula that shows the development of $$c_{d_w}$$ after the critical Mach number, $$M_{cr}$$, which is the number at which the flow becomes locally supersonic on the wing:

$$c_{d_w} = 20(M-M_{cr})^4 \tag{2}$$

If we apply the definition of $$(1)$$ to $$(2)$$ we obtain:

$$M_{cr} = M_{DD} - \left(\frac{0.1}{80}\right)^{1/3} \tag{3}$$

As you can see, the value of $$M_{DD}$$, is a little bit higher than the value of $$M_{cr}$$. First you get local super sonic flow, but the drag does not increase rapidly, as you further increase the Mach number, the drag starts to increase significantly, and you reach $$M_{DD}$$. See this image, where $$M_{DD}$$ is denoted by $$M_{CDR}$$.

Source

The value of $$M_{DD}$$ can be estimated using:

$$M_{DD} = \frac{K_A}{cos \Lambda} - \frac{t/c}{cos^2 \Lambda} -\frac{c_l}{10cos ^3\Lambda} \tag{4}$$

With

• $$\Lambda$$ is the sweep angle
• $$t/c$$ the relative thickness of the section lift coefficient
• $$c_l$$ the section lift coefficient
• $$K_A$$ the airfoil technology factor (a correction factor denoting the quality of the airfoil). Typical values for the airfoil technology factor are 0.87 for an NACA 6-series airfoil section or 0.95 for a supercritical section.

So the workflow is as follows:

1. Estimate $$M_{DD}$$ using $$(4)$$
2. Derive $$M_{cr}$$ using $$(3)$$
3. Determine the development of $$c_{d_w}$$ using $$(2)$$.

$$^1$$ Hilton, W.F., High Speed Aerodynamics, Longmans, Green & Co., London, 1952, pp. 47-49