Flow Around a Cylinder with Circulation

    The flow around a cylinder with circulation is
        Overscript[v, ⇀] = U(1 - R^2 (x^2 - y^2)/(x^2 + y^2)^2, R (-2x y)/(x^2 + y^2)^2) + γ/(2π) (-y/(x^2 + y^2), x/(x^2 + y^2))  defined for (x^2 + y^2)^(1/2) ≥R,
while the corresponding pressure has the formidable expression
        p = -(4 π^2 U^2 (R^4 - 2 R^2 (x^2 - y^2) + (x^2 + y^2)^2) - 4 π U y (R^2 + x^2 + y^2) γ + (x^2 + y^2) γ^2)/(8 π^2 (x^2 + y^2)^2).
We can then plot the pressure field; we begin with the case when γ = 4, R = 1, and U = 1/2.

Off[Plot3D :: "plnc"] ;            ... ;                

RowBox[{RowBox[{Cell[p(x,y) = ]}], , -(4 π^2 U^2 (R^4 - 2 R^2 (x^2 - y^2) + (x^2  ... 2)^2) - 4 π U y (R^2 + x^2 + y^2) γ + (x^2 + y^2) γ^2)/(8 π^2 (x^2 + y^2)^2)}]

[Graphics:../HTMLFiles/index_442.gif]

    We immediately notice that the symmetry of the previous two cases has been destroyed. The area above the obstacle (y>0) is at high pressure, while the area below the obstacle (y<0) is at low pressure. Indeed, here is the corresponding vector field and potential field.

Needs["Graphics`PlotField`"] ; Needs["Graphics`ImplicitPlot`"] ; γ =  ... ge {{-3R, 3R}, {-3R, 3R}}}], ]}], ;}],         }]

[Graphics:../HTMLFiles/index_446.gif]

The area of low pressure below the obstacle is characterized by densely packed equipotential curves and high velocity fluid flow, while the high pressure area has slow flow and sparsely packed equipotential curves.

    We can also see what happens as we change the parameters. In particular, as we increase γ from γ = 4 to γ = 7, while keeping R = 1 and U = 1/2, we obtain the following pressure graph.

Off[Plot3D :: "plnc"] ;            ... ;                

[Graphics:../HTMLFiles/index_453.gif]

Although this is superficially similar to the previous graph, its scale obscures an important fact. Recall that this example has a stagnation point at x = 0, y = (7 + (49 - 4π^2)^(1/2))/(2π) ≈1.60519; here is the graph of the field and equipotential curves.

Needs["Graphics`PlotField`"] ; Needs["Graphics`ImplicitPlot`"] ; & ... ge {{-3R, 3R}, {-3R, 3R}}}], ]}], ;}],         }]

[Graphics:../HTMLFiles/index_457.gif]

Bernoulli's equation implies that the pressure will have a maximum at a stagnation point; graphing the pressure field near the stagnation point shows this.

In[1]:=

Clear[R, U, γ] ; v[x_, y_] := U {1 - R^2 (x^2 - y^2)/(x^2 + y^2)^2, R^2 (-2 x y)/(x^2 + y ... ;                

[Graphics:../HTMLFiles/index_459.gif]

    In both instances, examining the pressure fields, we clearly expect that there is a nonzero force on the obstacle, and this is indeed the case.

x[θ_] := R Cos[θ] ;           &nbs ... 0; RowBox[{Print, [, RowBox[{RowBox[{   F   , Cell[ =]}], ,, Assuming[R>0, Simplify[F]]}], ]}]

                ⇀ RowBox[{RowBox[{   F   , Cell[ =]}], , {0, -R U γ}}]

We find that the resulting force is Overscript[F, ⇀] = (0, -R U γ), so the obstacle is pushed downwards.

    This is a particular instance of a more general result, called the Kutta-Joukowski Theorem. Define the circulation Γ of a vector field Overscript[v, ⇀] around a body Ω with boundary curve C by FormBox[RowBox[{Γ, =, RowBox[{∲_C,  , RowBox[{Overscript[v, ⇀](x, y) · d, Overscript[r, ⇀], StyleBox[Cell[], DisplayFormula]}]}]}], TraditionalForm].Then, for any plane conservative, incompressible flow exterior to Ω with uniform velocity Overscript[U, ⇀] at infinity, the force Overscript[F, ⇀] on the body is Overscript[F, ⇀] = -| Overscript[U, ⇀] | Γ Overscript[n, ⇀], where Overscript[n, ⇀] is a unit normal vector perpendicular to Overscript[U, ⇀]. It is easy for us to check that the circulation of our vector field around the obstacle is R γ; indeed

x[θ_] := R Cos[θ] ;           &nbs ...  Evaluate the integral *)Print["Γ = ", Assuming[R>0, Simplify[Γ]]]

Γ = R γ