7.5. Forces on a Two-Dimensional Body

In Section 3 the drag and lift forces per unit length on a circular cylinder in steady ideal flow were found to be zero and ρUΓ, respectively, when the circulation is clockwise. These results are also valid for any object with an arbitrary non-circular cross section that does not vary perpendicular to the x-y plane.

Blasius Theorem

Consider a stationary object of this type with extent B perpendicular to the plane of the flow, and let D (drag) be the stream-wise (x) force component and L (lift) be cross-stream or lateral (y) force (per unit depth) exerted on the object by the surrounding fluid. Thus, from Newton’s third law, the total force applied to the fluid by the object is F = –B(De_x + Le_y). For steady irrotational constant-density flow, conservation of momentum (4.17) within a stationary control volume implies:

∫A∗ρu(u·n)dA=−∫A∗pndA+F.

(7.54)

If the control surface A∗ is chosen to coincide with the body surface and the body is not moving, then u·n=0 and the flux integral on the left in (7.54) is zero, so:

Dex+Ley=−1B∫A∗pndA.

(7.55)

If C is the contour of the body’s cross section, then dA = Bds where ds = e_xdx + e_ydy is an element of C and ds = [(dx)² + (dy)²]^1/2. By definition, n must have unit magnitude, must be perpendicular to ds, and must point outward from the control volume, so n = (e_xdy – e_ydx)/ds. Using these relationships for n and dA, (7.55) can be separated into force components:

Dex+Ley=−1B∮Cp(exdy−eydx)dsBds=(−∮Cpdy)ex+(∮Cpdx)ey,

(7.56)

to identify the contour integrals leading to D and L. Here, C must be traversed in the counterclockwise direction.

Now switch from the physical domain to the complex z-plane to make use of the complex potential. This switch is accomplished here by replacing ds with dz = dx + idy and exploiting the dichotomy between real and imaginary parts to keep track of horizontal and vertical components (see Figure 7.18). To achieve the desired final result, construct the complex force:

D−iL=(−∮Cpdy)−i(∮Cpdx)=−i∮Cp(dx−idy)=−i∮Cpdz∗,

(7.57)

where ∗ denotes a complex conjugate. The pressure is found from the Bernoulli equation (7.18):

p∞+12ρU2=p+12ρ(u2+v2)=p+12ρ(u−iv)(u+iv),

where p_∞ and U are the pressure and horizontal flow speed far from the body. Inserting this into (7.57) produces:

Figure 7.18 Elemental forces in a plane on a two-dimensional object. Here the elemental horizontal and vertical force components (per unit depth) are –pdy and +pdx, respectively.

D−iL=−i∮C[p∞+12ρU2−12ρ(u−iv)(u+iv)]dz∗.

(7.58)

The integral of the constant terms, p_∞ + ρU²/2, around a closed contour is zero. The body-surface velocity vector and the surface element dz = |dz|e^iθ are parallel, so (u + iv)dz∗ can be rewritten:

(u+iv)dz∗=[u2+v2]1/2eiθ|dz|e−iθ=[u2+v2]1/2e−iθ|dz|eiθ=(u−iv)dz=(dw/dz)dz,

(7.59)

where (7.45) has been used for the final equality. Thus, (7.58) reduces to:

D−iL=iρ2∮C(dwdz)2dz,

(7.60)

a result known as the Blasius theorem. It applies to any steady planar ideal flow. Interestingly, the integral need not be carried out along the contour of the body because the theory of complex variables allows any contour surrounding the body to be chosen provided there are no singularities in (dw/dz)² between the body and the contour chosen.

Kutta-Zhukhovsky Lift Theorem

The Blasius theorem can be readily applied to an arbitrary cross-section object around which there is circulation –Γ. The flow can be considered a superposition of a uniform stream and a set of singularities such as vortex, doublet, source, and sink.

As there are no singularities outside the body, we shall take the contour C in the Blasius theorem at a very large distance from the body. From large distances, all singularities appear to be located near the origin z = 0, so the complex potential on the contour C will be of the form:

w=Uz+qs2πlnz+iΓ2πlnz+d2πz+...

When U, q_s, Γ, and d are positive and real, the first term represents a uniform flow in the x-direction, the second term represents a net source of fluid, the third term represents a clockwise vortex, and the fourth term represents a doublet. Because the body contour is closed, there can be no net flux of fluid into the domain. The sinks must scavenge all the flow introduced by the sources, so q_s = 0. The Blasius theorem, (7.60), then becomes:

D−iL=iρ2∮C(U+iΓ2πz−d2πz2+...)2dz=iρ2∮C(U2+iUΓπ1z+(Udπ−Γ24π2)1z2+...)dz.

(7.61)

To evaluate the contour integral in (7.61), we simply have to find the coefficient of the term proportional to 1/z in the integrand. This coefficient is known as the residue at z = 0 and the residue theorem of complex variable theory states that the value of a contour integral like (7.61) is 2πi times the sum of the residues at all singularities inside C. Here, the only singularity is at z = 0, and its residue is iUΓ/π, so:

D−iL=iρ22πi(iUΓπ)=−iρUΓ,orD=0andL=ρUΓ.

(7.62)

Thus, there is no drag on an arbitrary-cross-section object in steady two-dimensional, irrotational constant-density flow, a more general statement of d’Alembert's paradox. Given that non-zero drag forces are an omnipresent fact of everyday life, this might seem to eliminate any practical utility for ideal flow. However, there are at least three reasons to avoid this presumption. First, ideal flow streamlines indicate what a real flow should look like to achieve minimum pressure drag. Lower drag on real objects is often realized when object-geometry changes are made or boundary-layer separation-control strategies are implemented that allow real-flow streamlines to better match their ideal-flow counterparts. Second, the predicted circulation-dependent force on the object perpendicular to the oncoming stream – the lift force, L = ρUΓ – is basically correct. The result (7.62) is called the Kutta-Zhukhovsky lift theorem, and it plays a fundamental role in aero- and hydrodynamics. As described in Chapter 14, the circulation developed by an air- or hydrofoil is nearly proportional to U, so L is nearly proportional to U². And third, the influence of viscosity in real fluid flows takes some time to develop, so impulsively started flows and rapidly oscillating flows (i.e., acoustic fluctuations) often follow ideal flow streamlines.

Example 7.5

Revise the Kutta-Zhukhovsky Lift Theorem results for the case when q_s ≠ 0.

Solution

Start from (7.60) and consider a closed contour far from the origin as was done to reach (7.62), but include the source-or-sink term in the complex potential:

dwdz=U+qs2πz+iΓ2πz−d2πz2+...,soD−iL=iρ2∮C(U2+U(qs+iΓ)πz+(Udπ+(qs+iΓ)24π2)1z2+...)dz.

 

Here, the residue is U(q_s + iΓ)/π, so:

D−iL=iρ22πi(U(qs+iΓ)π)=−ρUqs−iρUΓ,orD=−ρUqsandL=ρUΓ.

 

The lift force is the same as in (7.62), but there is thrust (negative drag) on a source (q_s > 0) and drag on a sink (q_s < 0). The sign of this drag result can be understood in simple terms as follows. The streamlines for a uniform stream and a positive source are shown in Figure 7.7. Imagine a control volume that surrounds the fluid that comes from the source, extends downstream of the source, and terminates via a vertical segment between upper and lower branches of the boundary streamline (approximately the shape of a backwards letter “D”). The streamlines exterior to this CV are essentially the same as those surrounding a rocket heading into the on-coming stream, which implies a thrust force on the CV. Similarly, when a sink replaces the source, the sink removes a portion of the on-coming stream, along with the its momentum, from flow field. Thus, the sink feels a drag force that pushes in the direction (downstream) of the fluid momentum the sink removed.