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I need an equation to solve the following: I have a solid cone wherein I place the pointed section horizontally. I then switch on a horizontal airflow across the cone so that the flow strikes the point first and then flows over and around the body of the cone. If the flow is strong enough, I imagine that the resulting conical shape of the flow after it clears the base will at some point come together again, due to ambient air pressure. If that is what actually happens, is there an equation that can take air pressure, air temperature, flow velocity, cone dimensions, etc., and tell me approximately how far from the base of the cone the air flow will come back together again? Please see my rudimentary drawing. Thanks FLOW AROUND CONE

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One of the great challenges of fluid mechanics is that such a problem is only solvable using purely theoretical calculations in extremely limited circumstances, and the accuracy of those calculations is impossible to predict without an experimental sense of the onset of turbulence/flow separation. However, with the aid of experiments, an equation for the flow reconnection distance can be constructed with dimensional analysis.

Consider the parameters in your problem as the following:

  1. Fluid density $\rho$, with units of $\frac{kg}{m^3}$.
  2. Ambient flow speed in the horizontal direction $U$, with units of $\frac{m}{s}$.
  3. Ambient flow pressure $p$, with units of $\frac{kg}{m\ s^2}$.
  4. Ambient fluid viscosity $\mu$, with units of $\frac{kg}{m\ s}$.
  5. Streamwise cone length $L$, with units of $m$.
  6. Transverse cone width $w$, with units of $m$.

Suppose, also, that some equation of state existed (which you may not know) such that the ambient temperature $T$ is some function of the ambient pressure and density; $T = f(p,\rho)$. In that sense, the $6$ parameters above are the only ones that can be changed in your system. If, given any such parameters, there is a unique flow reconnection distance, it must be a function of these $6$ parameters.

Because the output of the goal function $d = g(\rho,p,\mu,U,L,w)$, the flow reconnection distance, has units of length, we can describe it as depending on the following four independent combinations of the system parameters, all with length units: $$\frac{\mu}{\rho U},\ \frac{\mu U}{p},\ L,\ w$$ (These are not unique—you could come up with many functions of them that also have length units—but they are simple and convenient.)

After this, you could try to perform experiments where you independently vary each of these derived parameters to find an answer, and construct a curve to determine a numerical approximation of $d = g(\frac{\mu}{\rho U},\frac{\mu U}{p},L,w)$.

There are also some physical arguments you can make to narrow down the parameters in your system; for example, if the flow is incompressible, your function should be of the form $d = g(\frac{\mu}{\rho U},L,w)$, and if you operate in either very high or low flow speed regimes, you can even (shakily) argue that $d = g(L,w)$ only; an example of these arguments applied to pressure losses in pipes and lift/drag forces can be found in Chapters 10 and 13 of my text.

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    $\begingroup$ This is a nice explanation (+1). But have you chance to read LANDAU&LIFSCHITZ FLUID MECHANICS? There is a theory of the laminar and turbulent wake as well in this book. $\endgroup$ Commented Dec 7, 2021 at 16:00
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    $\begingroup$ Landau & Lifschitz's treatment of laminar and turbulent wakes is phenomenal, but requires additional understanding about the transition between laminar and turbulent wakes and what the conditions for flow separation are in order to create a fully comprehensive model of wake characteristics as a function of system parameters. Although such a thing can be approximated using standard boundary-layer theory (also covered by L&L or Schlichting's texts), I follow the philosophy that such models are always detailed "local" approximations, complementing a rough but "global" experimental picture. $\endgroup$ Commented Dec 7, 2021 at 19:24

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