선 박 해 양 유 체 역 학

(1)

선 박 해 양 유 체 역 학

M ARINE H YDRODYNAMICS

2019년 6월 28일

Suh, Jung-Chun 서 정 천

Seoul National Univ., Dept. NAOE

서울대학교 공과대학 조선해양공학과

(2)

5 POTENTIAL FLOWS

5.1 Euler Equations for Inviscid Flows . . . 196

5.1.1 General Aspects. . . 196

5.1.2 Conservation of Circulation: Kelvin’s Theorem . . . 199

5.1.3 Irrotational Flow and Velocity Potential . . . 204

5.2 Energy Equation: Bernoulli Equation . . . 205

5.2.1 Integration of Euler Equations for Inviscid Fluids. . . 205

5.2.2 Case 1: Inviscid Steady (possibly Rotational) Incompressible Flow along Streamline . . . 206

5.2.3 Case 2: Inviscid Unsteady Irrotational Incompressible Flow. . . 209

5.3 Boundary Conditions . . . 210

5.3.1 Kinematic Boundary Condition . . . 210

5.3.2 Dynamic Boundary Condition: Free Surface Condition . . . 212

5.4 Simple Potential Flows . . . 213

5.4.1 Irrotational Inviscid Flows: Potential Flows . . . 213

(3)

5.4.2 Techniques for Solving the Laplace Equation . . . 215

5.4.3 Source/Sink/Dipole/Vortex in Free Stream (Uniform Flow) . . . 216

5.4.4 Flow around a Circular Cylinder . . . 221

5.5 Stream Function . . . 223

5.5.1 Various Singularity and Simple Potential Flows. . . 225

5.6 Separation of Variables . . . 227

5.6.1 Fixed Bodies and Moving Bodies . . . 230

5.7 Method of Images . . . 231

5.8 Green’s Theorem and Distributions of Singularities . ²³⁶ 5.8.1 Scalar Identity. . . 236

5.9 Hydrodynamics Pressure Forces: Forces and Moments Acting on a Body . . . 239

5.10 Force on a Moving Body in an Unbounded Fluid . . . 242

5.10.1 Far-field Behavior of Velocity Potential . . . 242

5.10.2 Decomposition of Velocity Potential: 6 Degrees of Freedom. . . . 243

5.11 Added Mass . . . 245

5.11.1 General Properties of Added-Mass Coefficients . . . 246

5.11.2 Added Mass of Simple Forms . . . 249

5.12 Body-Mass Force: Equations of Motion . . . 251

5.1 Euler Equations for Inviscid Flows

5.1.1 General Aspects

• Navier-Stokes equations deleting the viscous term.

– Governing equations: Continuity equation and Euler equations

∇ · q = 0, ρDq

Dt = −∇p + F_B (5.1)

(4)

5.1 Euler Equations for Inviscid Flows 197

– In a tensor notation,

∂u_i

∂x_i = 0, ∂u_i

∂t + u_j ∂u_i

∂x_j = −1 ρ

∂p

∂x_i + 1

ρ F_i (5.2) – For the gravity forces applied as a body force, Euler equations are,

where x2 is in the vertical upward direction.

∂u_i

∂t + uj

∂u_i

∂x_j = −1 ρ

∂

∂x_i (p + ρ g x2) (5.3)

• Euler Equations in Rectangular Coordinates (x, y, z)

∂u

∂t + u∂u

∂x + v∂u

∂y + w∂u

∂z = −1 ρ

∂p

∂x + g_x (5.4)

∂v

∂t + u∂v

∂x + v∂v

∂y + w∂v

∂z = −1 ρ

∂p

∂y + g_y (5.5)

∂w

∂t + u∂w

∂x + v∂w

∂y + w∂w

∂z = −1 ρ

∂p

∂z + g_z (5.6)

• Euler Equations in Cylindrical Coordinates (r, θ, z)

∂ur

∂t + u_r∂ur

∂r + uθ

r

∂ur

∂θ + u_z∂ur

∂z − u²_θ

r = −1 ρ

∂p

∂r + g_r (5.7)

∂u_θ

∂t + ur

∂u_θ

∂r + u_θ r

∂u_θ

∂θ + uz

∂u_θ

∂z + u_ru_θ

r = −1 ρ

∂p

r ∂θ + gθ (5.8)

∂u_z

∂t + u_r∂u_z

∂r + u_θ r

∂u_r

∂θ + u_z∂u_z

∂z = −1 ρ

∂p

∂z + g_z (5.9)

• Schematic Method for Visualization of Inviscid Flows

– The flow between two closely spaced parallel plates (Hele-Shaw flows) is laminar and viscous in the direction of the local pressure gradient. The velocity components (u, w) is proportional to the pressure gradient (see the solution for the plane Poiseuille flow):

u = 1 2µ

∂p

∂xy(y − h), w = 1 2µ

∂p

∂z y(y − h) (5.10)

(5)

Figure 5.1 Schematic view of the Hele-Shaw flow for visualization of inviscid potential flow.

(From Homsy 2000)

– The height-averaged velocity vector is proportional to the gradient of a scalar pressure:

q = (u, w) = − h²

12µ∇p (5.11)

– The two-dimensional streamlines in the steady flow about an intro- duced body closely approximate those ofinviscid potential flow. ¹

1 Movie: Hele-Shaw flows

./mmfm_movies/sportscarcompare.mov ./mmfm_movies/modeltcompare.mov

./mmfm_movies/airfoil_heleshaw22.mov ./mmfm_movies/airfoil_heleshaw45.mov ./mmfm_movies/heleshaw_tip2.mov

(6)

5.1.2 Conservation of Circulation: Kelvin’s Theorem

5.1.2.1 Introduction

• Circulation: Integrated tangential velocity around any closed contour C in the fluid,

Γ = I

C

u_idx_i (5.12)

• Kelvin’s Theorem (Conservation of Circulation)

For an ideal fluid acted upon by conservation forces (e.g., gravity) the circulation is constant about any closed material contour moving with the fluid particles or with velocity q. The material contour is composed of the same fluid particles.

dΓ

dt = 0 (5.13)

• Acceleration of a flat plate in a kitchen sink.

• Stirring of a spoon in a coffee cup. ²

– If the motion is at right angle to the flow, vortices will be shed from both edges.

– If it is at a small angle of attack, only one vortex be shed from the trailing edge (T. E.).

5.1.2.2 Time rate of circulation for barotropic fluids

• In any flow of a barotropic inviscid fluid, the circulation about any closed path does not vary with time if the contour is imagined to move with the fluid, that is, always to be made up of the same particles.

• We give here a different proof, which offered more generality. We begin by considering the contour integral Γ =

I

C

q · d` where q is any vector quantity and C (material contour) is carried by the fluid.

2 Movie: Conservation of Circulation(Coffee cup) ./mmfm_movies/Spoon2.mov

(7)

• Now consider the time derivative of the circulation for a closed curve fol- lowing the motion:

dΓ dt = d

dt I

C(t)

q(x, t) · s(x, t)d` (5.14) where s(x, t) is the unit tangent vectors along the integration path of the contour C.

• This differential is similar to the starting point in our derivation of the Reynolds transport theorem in Chapter A, but for a moving curve instead of a moving volume. We make the same transformation from spatial (x) to initial coordinates (ξ):

dΓ dt =

I

C(0)

∂

∂t

q^∗(ξ, t) · dx^∗(ξ, t) d` d`

= I

C(0)

"

∂q^∗

∂t ξ

· dx^∗

d` + q^∗ · ∂

∂t

∂x^∗(ξ, t)

∂ξj

dξ_j d`

# d`

= I

C(0)

"

∂q^∗

∂t ξ

· dx^∗

d` + q^∗ ·

∂q^∗(ξ, t)

∂ξ_j

dξ_j d`

# d`

= I

C(t)

Dq

Dt · s + q ·(s · ∇)q

d`

= I

C(t)

Dq

Dt · s + ∂

∂`

1 2q²

d`

= I

C(t)

Dq

Dt · s d` (5.15)

• Now the total (material) derivative term is the LHS of the momentum equation allowing us to put the RHS of the momentum equation into the integral:

dΓ dt =

I

C(t)

−∇p

ρ + F_B

ρ + ν∇²q

· d` (5.16)

• For barotropic fluids and conservative body forces, the time rate of change of the circulation reduces to an integral containing only viscous terms since

(8)

the integral of a gradient about a closed curve is zero:

dΓ dt = ν

I

C(t)

∇²q · d` (5.17)

• Every flow that can be produced (without friction) in a barotropic fluid initially at rest, initially in a uniform stream, or initially in any irrotational state, must be an irrotational flow. ³

5.1.2.3 Case of inviscid flows

• Thus when either the kinematic viscosity or the gradient of the vorticity is small, the circulation will be preserved about a curve moving with the fluid.

• Note that the circulation is not influenced by the pressure or conservative body forces and that if the fluid were inviscid the equation is irrespective of the field vorticity. For such flows,

DΓ

Dt = 0 (5.19)

as the theorem states.

• Applying this to the contours that enclose vortex filaments, we see immediately that such vortices do not vary in strength as they move about in any barotropic inviscid fluid. From the theorem above, it becomes clear that if a fluid particle in this type of fluid once has zero vorticity it will always have zero vorticity.

• For consider a contour surrounding a very small sample of fluid; if ω is zero for this sample, Γ is also zero, and according to the theorem must remain

3If the fluid is baroclinic, the circulation can be modified because of the baroclinic generation of vorticity as dΓ

dt = ν I

C(t)

∇²q · d` + Z

S

1

ρ²(∇ρ × ∇p) · d` (5.18)

where the surface integral on the right-hand side is performed over the area bounded by material line. For details, see Milne-Thomson, L. M. (1968), Theoretical Hydrodynamics, 5th edition, Macmillan, London, p. 84.

(9)

so. But this certainly implies that ω remains zero, since the statement is true for every contour that surrounds any part of the sample. ⁴

5.1.2.4 Another derivation of Kelvin’s theorem

• Derivative of circulation with respect to time:

dΓ dt = d

dt I

C

uidxi = I

C

∂

∂t + uj

∂

∂x_j

(uidxi) (5.20) where d/dt is the material derivative.

• A finite sum dΓ

dt = lim

N →∞

X

n

∂

∂t + u_j ∂

∂xj

u⁽ⁿ⁾_i δx⁽ⁿ⁾_i

(5.21)

where δx⁽ⁿ⁾_i ≡ x⁽ⁿ⁺¹⁾_i − x⁽ⁿ⁾_i → 0 and u⁽ⁿ⁾_i = ∂x⁽ⁿ⁾_i

∂t .

• Noting that the coordinates depend on time not on the space coordinates, dΓ

dt = lim

N →∞

X

n

∂u_i

∂t + uj

∂u_i

∂x_j

x⁽ⁿ⁺¹⁾_i − x⁽ⁿ⁾_i + ui

u⁽ⁿ⁺¹⁾_i − u⁽ⁿ⁾_i

= I

C

∂u_i

∂t + uj

∂u_i

∂x_j

dxi+ I

C

uidui (5.22)

• Integral of a perfect differential over a closed contour dΓ

dt = −1 ρ

I

C

∂

∂xi

(p + ρ g x₂) + I

C

1

2d(u_iu_i) (5.23)

• Perfect differential with identical limits

− 1 ρ

p + ρ g x₂ − 1 2u_j u_j

B A

(5.24)

4See Kuethe, A. M. and Chow, C.-Y. (1976), Foundations of Aerodynamics: Bases of Aerodynamic Design, Wiley, pp. 53–54.

(10)

becomes 0. This results in

DΓ

Dt = 0. (5.25)

as the theorem states.

5.1.2.5 Circulation generation: Kutta condition

• Potential flow without circulation

• Thin boundary layer: high Reynolds no., thin and streamlined body

• Smooth tangential flow at T.E.(Trailing Edge)

• Circulation around foil moves stagnation point back at T.E.

Figure 5.2 Initial development of the circulatory flow past a hydrofoil by Kutta condition.

(From Newman 1977; MIT website 2004)

• For steady-state hydrofoils, the starting vortex can be disregarded, since it will be far downstream and dissipated by viscous diffusion.

– Net circulation about the foil by the smooth local flow at the trailing edge is essential to the development of the desired lift force.

– This assumption is imposed actually by Kutta condition requiring the velocity at the T. E. to be finite.

(11)

5.1.3 Irrotational Flow and Velocity Potential

• In regions where the flow is irrotational, the line integral around an entire closed path is the circulation and is zero because the flow is irrotational.

5 This implies that the open line integral Z

q · d` is independent of the path within the regions, but only dependent of the end points of the path.

Therefore, choosing A as a fixed point and B as a varying point,

Figure 5.3 Open and closed line integrals in region of an irrational flow: Path independent integrals of velocity potential. (From Sears 1970)

Z B(x,y,z) A

q · d` = φ(x, y, z), (5.26) and

dφ = q · d` (5.27)

• Now we see that dφ = d` · ∇φ from Eq. (A.66) and hence d` · ∇φ = q · d`

for arbitrary choice of d`. This means that q = ∇φ, or ui = ∂φ

∂x_i in tensor notation (5.28)

• By retracing these steps you will see immediately that this result has noth- ing to do with the physical meaning of q. That is, the result Eq. (5.28) will

5 Movie: Irrotational flow ./mmfm_movies/3_14040.mov

(12)

5.2 Energy Equation: Bernoulli Equation 205

follow for every vector function q whose curl is zero. Moreover, the condition ∇ × q = 0 is necessary, as well as sufficient, for the result q = ∇φ, because the curl of every gradient is identically zero.

• In the case considered here, where q(x, t) is the fluid velocity, φ(x, t) is called the velocity potential. The surfaces φ = constant are called equipo- tentialsurface; thus q is the vector perpendicular to these surfaces at every point, and its magnitude is that of derivative ∂φ/∂n in the normal direction. These statements are verified by using the relation dφ = d` · ∇φ.

• Since the equation of continuity is ∇·q and q is the gradient of the velocity potential φ, the differential equation satisfied by φ is Laplace equation:

∇ · (∇φ) = ∇²φ = 0 (5.29)

or

∂

∂xi

∂φ

∂xi

= ∂²φ

∂xi∂xi

= 0, ∇²φ ≡ ∂²φ

∂x² + ∂²φ

∂y² + ∂²φ

∂z² = 0 (5.30)

5.2 Energy Equation: Bernoulli Equation

5.2.1 Integration of Euler Equations for Inviscid Fluids

• The equations of motion for inviscid fluids are called Euler’s equations, with dropping the last term of Eq. (4.37).

• The term q · ∇q, which occurs in the equations can be transformed by the vector expansion formula:

∇ q² = ∇(q · q) = 2q · ∇q + 2q × (∇ × q) (5.31) Thus

q · ∇q = 1

2∇ q² − q × ω (5.32)

(13)

• Eventually an alternate form of the Euler’s equation is

∂q

∂t + ω × q = −1

ρ∇p + 1

ρF_B − ∇ q² 2

(5.33)

• It is often assumed that the body force F_B is derivable from a potential;

that is, that it is a conservative force, such as gravity. Then we can write F_B = −∇Ω, and the equations appear in

∂q

∂t − q × ω = −∇ q² 2 + Ω

− 1

ρ∇p (5.34)

5.2.2 Case 1: Inviscid Steady (possibly Rotational) Incompressible Flow along Streamline

• Consider Euler’s equation u_j ∂ui

∂u_j = − ∂

∂x_i

p

ρ + g x₂

(5.35)

• Inner product with the velocity uiuj

∂u_i

∂u_j = −ui

∂

∂x_i

p

ρ + g x2

(5.36)

• Left side of this equation can be written as u_iu_j ∂ui

∂u_j = 1 2u_j ∂

∂u_j (u_iu_i) = 1 2u_i ∂

∂u_i (u_ju_j) (5.37)

• Collecting terms on one side u_i ∂

∂x_i

p ρ + 1

2u_ju_j + g x₂

= 0 (5.38)

• Along streamline, the quantity in parentheses is constant : p + 1

2ρ q² + ρ g x₂ = C (5.39)

(14)

where q² = u_iu_i = u² + v² + w².

• If the vertical coordinate x2 is replaced by z, it becomes p

ρ + 1

2q² + g z = Constant (5.40)

5.2.2.1 Steady barotropic flows

• This is about as far as we can go with complete generality, but the equations can be simplified still further if the fluid is barotropic; that is, if the density ρ depends on the pressure p only: ρ = ρ(p). ⁶

• Example of this state of affairs are compressible fluids flowing adiabati- cally (p ∼ ρ^k) or isothermally (p ∼ ρ), or, of course, incompressible fluids (ρ = constant). For this case we need not assume irrotational flow.

– In these cases the term 1

ρ∇p can also be expressed as the gradient of a function, for consider

d`· ∇p

ρ(p) = dp ρ(p) = d

Z dp

ρ(p) = d` · ∇

Z dp

ρ(p) (5.41)

– Since d` is arbitrary, ∇p

ρ(p) = ∇

Z dp

ρ(p), and the Euler’s equations of motion are reduced to

∂q

∂t − q × ω = −∇ q²

2 + Ω +

Z dp ρ

(5.42)

• For steady flow, the Euler equations then read, in the original form, q · ∇q = −∇

Z dp ρ + Ω

(5.43)

• We shall now show that this can be integrated along individual streamlines;

that is, we shall obtain an integral that will tell how the quantities behave along a streamline, but not how they change from streamline to streamline.

6We call a fluid baroclinic if the density does not depend on the pressure.

(15)

• Let an orthogonal curvilinear coordinate system be defined so that s is measured along a streamline, and r and t normal to it. Then q = (q, 0, 0), and q · ∇q =

q ∂q

∂s, · · · , · · ·

(as may be verified by reference to the formulas in general curvilinear orthogonal coordinates).

• Let us substitute this into Eq. (5.43) and then multiply both sides by ·ds:

q ∂q

∂s ds = − ∂

∂s

Z dp ρ + Ω

ds (5.44)

and, integrating along the streamline 1

2q² +

Z dp

ρ + Ω = Cs (5.45)

where Cs is the constant of integration, and we give it the subscript s to emphasize that the constant may vary from streamline to streamline.

• Since Eqs. (5.56) and (5.45) must yield the same result in cases of steady, irrotational barotropic flow, we see that the irrotational assumption is equivalent to taking the same constant, C_s, for all streamlines:

1 2q² +

Z dp

ρ + Ω = constant (5.46)

and finally if this is also incompressible, 1

2ρ q² + p + ρ Ω = constant (5.47)

• This will be recognized as Bernoulli’s equation, which is an energy equation, although we obtained it by integration of momentum equations. In fact, Eqs. (5.56), (5.57), and (5.43) are also sometimes called generalized forms of Bernoulli’s equation.

• The corresponding Bernoulli equation, when the gravity for body force is applied,

Z dp ρ + 1

2q² + g z = Constant (5.48)

(16)

5.2.3 Case 2: Inviscid Unsteady Irrotational Incompressible Flow

• With the velocity potential, the alternative form

∂

∂t

∂φ

∂x_i

+ ∂φ

∂x_j

∂

∂x_j

∂φ

∂x_i

= − ∂

∂x_i

p

ρ + g x₂

(5.49)

• Direct differentiation

∂φ

∂xj

∂

∂xj

∂φ

∂xi

= 1 2

∂

∂xi

∂φ

∂xj

∂φ

∂xj

(5.50)

• Perfect differential

∂

∂xi

∂φ

∂t + 1 2

∂φ

∂xj

∂φ

∂xj

= − ∂

∂xi

p

ρ + g x₂

(5.51)

• Integrating with respect to the space variables:

∂φ

∂t + 1 2

∂φ

∂x_j

∂φ

∂x_j

= − p

ρ + g x₂

+ C(t) (5.52) where ‘C(t)’ can be absorbed into velocity potential

φ⁰ = φ − Z

C(t) dt (5.53)

• Consequently,

p

ρ + ∂φ

∂t + 1

2q² + g x₂ = C(t) (5.54)

5.2.3.1 Irrotational barotropic flows

• In this type of flow q is ∇φ. The left-hand side of Eq. (5.42) becomes simply ∂

∂t (∇φ), and since the time and space derivatives are independent

(17)

and can be exchanged in order, this is equal to ∇ ∂φ

∂t

. Thus

∇ ∂φ

∂t + q²

2 + Ω +

Z dp ρ

= 0 (5.55)

• But when the gradient of a function is zero throughout a region, the function must certainly be constant throughout the region– or rather, since the gradient involves space derivatives only, the function must be constant throughout the region at any instant, but may vary with time.

• The integrated form is therefore

∂φ

∂t + q²

2 + Ω +

Z dp

ρ = C(t) (5.56)

• Remember that the term Z

dp/ρ is just a function of ρ (or p) whose form is known as soon as the particular barotropic law ρ = ρ(p) is specified.

• For example, the simplest law is that of the incompressible fluid: ρ =constant.

Hence the integrated equation for incompressible, frictionless, irrotational, unsteady flow is

∂φ

∂t + q²

2 + Ω + p

ρ = C(t) (5.57)

5.3 Boundary Conditions

5.3.1 Kinematic Boundary Condition

• In order to determine the velocity field of a potential flow, we need the boundary condition for velocity on the body surface. For general formulation, we consider a moving boundary here.

• Now x = x(ξ, t) denotes the position vector of a point on the moving surface, where we take ξ as the initial position vector of the point.

(18)

5.3 Boundary Conditions 211

• The velocity of the moving surface is then u = ∂x

∂t ξ

(5.58)

• The boundary condition on the moving boundary is that the normal component of the fluid velocity must equal the normal velocity of the moving boundary:

u · n = U_B · n. (5.59)

5.3.1.1 Alternate form

• When the moving boundary is specified by a function, F (x, t) = 0, the boundary condition can be written in an alternate form.

• Along the path of motion x = x(ξ, t) and the moving surface F x(ξ, t), t = 0, the particles are always located at the material surface: ∂F

∂t ξ

= 0. It implies that

∂F

∂t + ∂F

∂xi

∂x_i

∂t

ξ

= 0 (5.60)

or ∂F

∂t + U_B · ∇F = 0 (5.61)

• The normal vector is defined from the function F as⁷ n = ∇F

|∇F | (5.62)

Then,

∂F

∂t + (U_B · n) |∇F | = 0 (5.63)

7For scalar field F (x), we consider a curve x(σ) on a surface of F (x) = constant, which is specified with the parameter σ. Then dF (x)

dσ =dx(σ)

dσ · ∇F (x) Because F is constant along the curve x(σ), dF

dσ = 0. Since dx(σ)

dσ is tangent to x(σ), it requires either that ∇F = 0 or that ∇F is perpendicular todx(σ)

dσ (namely, to x(σ)).

Therefore non-zero ∇F is perpendicular to the surface F = constant.

(19)

• Use the condition for the normal components of the velocities of both fluid and surface u · n = U_B · n, to obtain

∂F

∂t + u · n |∇F | = 0. (5.64)

• And we find

∂F

∂t + u · ∇F = 0 (5.65)

or

DF

Dt = 0. (5.66)

• This expression is valid for all material surfaces and for any flow conditions, e.g., for unsteady compressible viscous fluids. If F is independent of time, the expression reduces to the simple one: u · n = 0.

• But, if we use a relative coordinate system fixed to a moving body to de- scribe the flow field, the influence of the frame velocity of the moving coordinate system should be added. The detailed formulation is given in Chapter A.

5.3.2 Dynamic Boundary Condition: Free Surface Condition

• For inviscid fluids in the absence of surface tension, the pressure is contin- uous across any interface between two fluids.

• Perhaps the most familiar case is the liquid free surface under the atmospheric air. The boundary condition is that in the liquid just below the free surface, the pressure is the same as in the air just above the free surface.

The atmospheric pressure p_a can generally be taken as a constant.

• For a liquid free surface that is at rest far ahead of a body whose motion creates disturbances on the free surface, the free surface boundary conditions are x₃ = ζ(x₁, x₂, t) and p(x1, x₂, ζ, t) = p_a.

• The first condition is the kinematic condition which describes the free surface height from x3 = 0 and the second one is dynamic for imposing the

(20)

5.4 Simple Potential Flows 213

atmospheric pressure on the free surface.

• For upstream we can expect disturbance-free condition. Hence Bernoulli’s equation on the free surface reduces to the expression:

(p − p_a)

ρ + gζ(x1, x2, t) + 1

2(∇φ)² + ∂φ

∂t = 0 (5.67)

• With p = pa in this equation, we have a suitable form to derive explicitly the free surface height if the velocity potential φ is known:

ζ(x₁, x₂, t) = −1 g

∂φ

∂t + 1

2(∇φ)²

(5.68)

5.4 Simple Potential Flows

5.4.1 Irrotational Inviscid Flows: Potential Flows

• We shall devote considerable attention to the study of irrotational motions of incompressible fluids. As we have seen, the irrotational approximation is likely to be valid throughout much of the flow.

• Since the equation of continuity is ∇·q and q is the gradient of the velocity potential φ, the differential equation satisfied by φ is Laplace equation:

∇ · (∇φ) = ∇²φ = 0 (5.69)

• The pressure-velocity relation (bernoulli equation) is given by the integrated dynamical equation:

∂φ

∂t + 1

2q² + p

ρ + Ω = C(t) (5.70)

• In a more general case, even if we restrict ourselves to barotropic fluids, we have five dependent variables: u, v, w, p, ρ; and five equations to solve for them:

(21)

3 equations of motion, 1 equation of continuity, and 1 equation of state:

ρ = p(ρ)

• We see now that an extreme simplification has been achieved in the irrotational incompressible case, for the equation of state has degenerated to ρ = constant, and we have replaced u, v, w by the velocity potential φ, leaving only two unknowns (φ and p) and two equations, Eqs. (5.69) and (5.70).

• Moreover, Eq. (5.70) has been integrated, and constitutes a formula for calculation of p when Eq. (5.69) has been solved. The only mathematical problem that remains is the solution of Laplace’s equation Eq. (5.69), with the appropriate boundary conditions.

• The most surprising result is that the dynamical equations do not impose any restrictions on the flow. Any solution of the equation of continuity Eq. (5.69) is a possible flow pattern, for some set of boundary conditions.

Another statement of this situation is that everykinematically possible flow is dynamically possible.

• It is also important to notice that Eq. (5.69) does not involve t. In a case of unsteady flow, the boundary conditions will vary with time. All that is re- quired is that we solve Laplace’s equation with the instantaneous boundary conditions. Another statement of this is that every unsteady flow pattern is a possible steady flow pattern (and vice versa). Of course, the corresponding pressure will depend on whether the flow is steady or not.

• If f satisfies Laplace’s equation in a region, then f has no maxima or minima in that region. For any volume V , enclosed in a surface S, lying entirely inside the region:

0 = Z

V

∇²f dV = Z

V

∇ · ∇f dV = Z

S

n · ∇f dS = Z

S

∂f

∂n dS (5.71) But if f has a maxima at any point P , we can surely obtain a negative value of

Z

S

(∂f /∂n) dS by taking V to enclose P and making it small enough. Similarly we can obtain a positive value by integrating ∂f

∂n around a minimum. Consequently there can be neither.

(22)

5.4.1.1 Uniqueness of Laplace equation

• Let us consider the uniqueness of the Laplace equation. Suppose two solutions φ1 and φ2 which have the same value of ∇²φ in V and the same value of φ or n · ∇φ on S, then we could construct a third solution φ⁰⁰ ≡ φ₁− φ₂ which had ∇²φ⁰⁰ = 0 in V , and either φ⁰⁰ = 0 or n · ∇φ⁰⁰ = 0 on S.

Z

V

φ⁰⁰∇²φ⁰⁰ + ∇φ⁰⁰· ∇φ⁰⁰ dV = I

S

[φ⁰⁰n · ∇φ⁰⁰] dS = 0 (5.72) and this reduces to only

Z

V

(∇φ⁰⁰ · ∇φ⁰⁰) dV = 0 (5.73)

• Since (∇φ)² is always greater than or equal to zero (i.e., the integrand is

‘positive definite’), the only solution is

∇φ⁰⁰ · ∇φ⁰⁰ = 0 (5.74)

• This requires that φ⁰⁰ be at most a constant. If φ were specified on the boundary, the constant is zero. If n · ∇φ is specified on the boundary, φ is uniquely determined by the integral to within a constant. It is important to recognize that our expression for φ is in terms of φ and n · ∇φ and the above consideration shows we need specify only one of these on the boundary.

• Also we have assumed that the field boundaries are fixed. If they were to depend on the field, then special conditions must be specified to insure the solution is unique. In addition to this uniqueness, we should also consider the far-field behavior of φ as the distance r goes to infinity. ⁸

5.4.2 Techniques for Solving the Laplace Equation (a) Separation of variables

8Detailed consideration is found in Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge.

(23)

– Standard technique limited to systems for which separation can pro- duce a solution to the Laplace equation.

– It is desirable that the boundary condition can be easily applied.

– Usually the value of one coordinate fixed describes the surface.

(b) Superposition

– Superposition of elementary solutions of the Laplace equation to sat- isfy the boundary condition.

– The distributions can be specified to be anywhere inside the body with elementary solutions employing singularity distributions (source, sink, vortex, dipole, etc.) entirely within the body.

(c) Identity

– Identities relating surface distributions of various singularities to field values of the quantity of interest.

– We have defined two different identities of interest in the vector anal- ysis review:

(i) Green’s second identity for scalar and (ii) an identity for an arbitrary vector.

Note: The items (b) and (c) differ in the location of the singularity but not in principle, (e.g., lifting surface methods vs. panel methods.)

5.4.3 Source/Sink/Dipole/Vortex in Free Stream (Uniform Flow)

• Velocity potential of a point source at the origin φ = −m

4π

1

px² + y² + z² in 3-dimensions (5.75)

φ = m

2π log_ep

x² + y² in 2-dimensions (5.76)

(24)

• Half-Body: Source + Free Stream φ = U x − m

4π

1

px² + y² + z² (5.77)

Figure 5.4 Semi-infinite half-body generated by a point source at the origin in free stream.

(From Newman 1977)

– Stagnation point: x_s = −

r m

4πU.

• Rankine Ovoid: Free Steam + Source + Sink φ = U x − m

4π

1

p(x + a)² + y² + z² + m 4π

1

p(x − a)² + y² + z² (5.78)

– Body length (l) between the stagnation points:

l² 4 − a²

2

= 2 a l

m 4πU

(5.79) – Half-breadth of body (b) (circle radius of mid-plane):

m = 2π Z b

0

"

U + 2am/4π (a² + R²)^3/2

#

R dR = π U b² r

1 + b²

a² (5.80)

(25)

Figure 5.5 Rankine ovoid generated by a point source and sink combination in free stream.

(From Newman 1977)

• Dipole (or Doublet)

– The separation 2a between source and sink shrinks to zero.

– To preserve a finite effect from the singularities as they are brought together, their strengths m must increase at the same rate.

– It is necessary to make the product µ = 2ma a constant, φ = lim

a→0

µ 8πa

"

− 1

p(x + a)² + y² + z² + 1

p(x − a)² + y² + z²

#

= µ

4π

∂

∂a

"

1

p(x − a)² + y² + z²

#

a=0

= µx

4π (x² + y² + z²)^3/2 (5.81)

– The corresponding potential for two-dimensional dipole φ = µ

2π x

x² + y² (5.82)

• Sphere (or Circle): Dipole + Free Stream – In 3-Dimesion (Sphere)

φ = U x + µx

4π (x² + y² + z²)^3/2 (5.83)

(26)

Figure 5.6 Schematic concept of dipole singularity by approach of source and sink. (From Parsons 1984)

Figure 5.7 Spherical coordinate system for showing the flow about a sphere.

(27)

– In a Spherical Coordinate System

φ = U r cos θ + µ cos θ

4πr² (5.84)

Here the sphere radius R =

µ 2πU

1/3

.

• In 2-Dimesion (Circular cylinder)

φ = U x + µx

2π (x² + y²) (5.85)

In polar coordinate (r, θ),

φ = U r cos θ + µ cos θ

2πr (5.86)

Here the cylinder radius R =

µ 2πU

1/2

.

5.4.3.1 Flow past a sphere

• Let us first consider the steady irrotational flow past a sphere. After the doublet strength µ is replaced in terms of the stream speed U and the radius R of the sphere, as the student can easily verify, the flow is described by

φ = 1 2U

2r + R³ r²

cos θ, ψ = 1 2U

r² − R³ r

sin²θ, qr = U

1 − R³ r³

cos θ, qθ = −1 2U

2 + R³ r³

sin θ











(5.87)

• On the sphere, the velocity is given by q_θ = −3

2U sin θ (5.88)

so that there are two stagnation points at θ = 0, π, and the maximum local speed is 50% greater than the stream speed.

(28)

• The pressure is given by

p − p₀ 1 2ρ U²

= 1 − 9

4sin²θ (5.89)

Figure 5.8 Pressure distribution over the surface of a sphere. (From Parsons 1984)

Now, in addition to the axial symmetry we see that the velocities and pressure are distributed symmetrically with respect to the equatorial plane θ = π/2. Consequently there can be no force on the sphere.

• In describing real-fluid flows, the boundary layer separates from the surface just forward of the equator θ = π/2 (in agreement with viscous-fluid theory), and from there back the flow loses its resemblance to perfect-fluid flow. It is clear that the boundary-layer separation is ultimately responsible for the appreciable drag of the sphere.

5.4.4 Flow around a Circular Cylinder

• We shall proceed to the consideration of the plane steady irrotational flow around a circular cylinder. ⁹ In terms of the radius of the cylinder, R, the

9 Movie: Potential flow about a circle ./mmfm_movies/Circle1.mov

(29)

formulas for this case are φ = U

r + R² r

cos θ, ψ = U

r − R² r

sin θ, q_r = U

1 − R² r²

cos θ, q_θ = −U

1 + R² r²

sin θ











(5.90)

• Thus the maximum surface speed is 2U in this case, and the local pressures correspondingly lower than on the sphere. Once more the pressure is distributed symmetrically fore-and-aft, and there is no force on the cylinder.

5.4.4.1 Circular cylinder with circulation

• In this case, we encounter a new and important phenomenon. The boundary conditions satisfied by equation Eq. (5.90) are

qr = 0 when r = R, and q → U i as r → ∞. (5.91)

• But these would be just as well satisfied if we were to superimpose a plane vortex flow about the origin, of any desired strength; namely,

φ = U

r + R² r

cos θ + Γ

2πθ for 0 ≤ θ < 2π (5.92) ψ = U

r − R² r

sin θ − Γ

2π ln r (5.93)

q_r = U

1 − R² r²

cos θ (5.94)

qθ = −U

1 + R² r²

sin θ + Γ

2πr (5.95)

• With circulation, the local velocity at the surface becomes −2U sin θ + Γ/2πR. Thus the stagnation points have moved to

θ = sin⁻¹ Γ

4πU R (5.96)

(30)

5.5 Stream Function 223

provided that |Γ| ≤ 4πU R. (If |Γ| has a greater value, the stagnation points merge and occur in the flow outside the cylinder.)

• The fluid pressure on the cylinder is now p = p₀ + 1

2ρ U² − 1 2ρ

2 U sin θ − Γ 2πR

2

(5.97) and again, we see that there is no force component in the x-direction, i.e.

no drag.

• The force component in the y direction is easily computed:

Y = − Z 2π

0

p R sin θ dθ = −ρ U Γ (5.98)

• We see that there is lift on a circular cylinder with circulation. This is obviously related to the so-called ‘Magnus effect’, which produces lift on a rotating cylinder in a real fluid, the circulation is produced by the action of viscosity near the spinning cylinder.

5.5 Stream Function

• Two-dimensional flow

– Satisfying the continuity equation ∂u

∂x + ∂v

∂y = 0 automatically, the stream function ψ is defined by q = ∇ × Ψ, where we have only a component in 2-D, Ψ = (0, 0, ψ), so that

u = ∂ψ

∂y, v = −∂ψ

∂x (5.99)

– In 2-D polar coordinates, the continuity equation and the stream func-

(31)

tion ψ(r, θ):

∂ (r u_r)

∂r + ∂u_θ

∂θ = 0 (5.100)

u_r = 1 r

∂ψ

∂θ and u_θ = −∂ψ

∂r (5.101)

• Axi-symmetric flow that is independent of θ,

– Stream function ψ, has only a component in the θ-direction, satisfying the continuity equation automatically.

q = ∇ × Ψ with Ψ = (0, ψ/R, 0) (5.102) uR = −1

R

∂ψ

∂x, and ux = 1 R

∂ψ

∂R (5.103)

• Measure of flux across a contour C

Figure 5.9 Measure of flux across a contour: Stream function. (From Newman 1977; MIT website 2004)

– 2-Dimensional case

ψ = Z

C

(u dy − v dx) (5.104)

– Axi-Symmetric case Q = 2π

Z

S(R,x)

(u_xR dR − u_RR dx) = 2πψ (5.105)

• Example of Axi-Symmetric Flow: Rankine Ovoid¹⁰

10 Movie: Potential flow about a Rankine ovoid

(32)

5.5 Stream Function 225

– The velocity potential in cylindrical coordinates, φ(R, θ, x) = U x−m

4π (x + a)² + R²−1/2

+ m

4π (x − a)² + R²−1/2

(5.106) – The stream function as a measure of flux at the plane of x = constant:

ψ(R, x) = Z R

0

∂φ

∂xR⁰dR⁰ = ∂

∂x Z R

0

φR⁰dR⁰

= ∂

∂x

1

2U xR² − m

4π (x + a)² + R²1/2

+m

4π (x − a)² + R²1/2o

= 1

2U R² − m(x + a) 4π [(x + a)² + R²]^1/2

+ m(x − a)

4π [(x − a)² + R²]^1/2 (5.107) – Equation of the body surface: Equation of dividing streamlines

ψ = 0 = 2πU R²

m − (x + a)

[(x + a)² + R²]^1/2

+ (x − a)

[(x − a)² + R²]^1/2 (5.108)

5.5.1 Various Singularity and Simple Potential Flows

• Velocity Potential vs. Stream Function

Table 5.1 Velocity potential versus stream function Velocity Potential Stream Function

Definition q = ∇φ q = ∇ × Ψ

Continuity Eq.

∇ · q = 0 ∇²φ = 0 Automatically satisfied

Irrotationality ∇ × (∇ × Ψ) = ∇(∇ · Ψ) − ∇²Ψ = 0

∇ × q = 0 Automatically satisfied (2-D case: ∇²ψ = 0)

./mmfm_movies/Heleshaw.mov

(33)

Figure 5.10 Streamlines about a Rankine ovoid. (From Newman 1977; Parsons 1984)

(34)

5.6 Separation of Variables 227

Table 5.2 Various singularity and simple potential flows

Singularity Velocity Potential Stream Function

Position at x₀ φ(r, θ, z) in Cylindrical Coordinates ψ(r, θ) in Polar Coordinates φ(x, y, z) in Cartersian Coordinates ψ(x, y) in Cartersian Coordinates

Uniform flow U r cos θ + V r sin θ + W z U r sin θ − V r cos θ

Velocity (U, V, W ) U x + V y + W z V y − U x

2-D Source m

2πlog_e r m

2πθ

Strength m m

2πlog_ep

(x − x0)²+ (y − y0)² m

2πtan⁻¹ y − y0

x − x₀

3-D Source −m

4π 1

r —

Strength m −m

4π

1

p(x − x₀)²+ (y − y₀)²+ (z − z₀)² —

2-D Dipole − µ

2π

cos θ cos α + sin θ sin α r

µ 2π

sin θ cos α + cos θ sin α r

Strength µ

Direction: angle α − µ 2π

(x − x₀) cos α + (y − y₀) sin α (x − x₀)²+ (y − y₀)²

µ 2π

(y − y₀) cos α + (x − x₀) sin α (x − x₀)²+ (y − y₀)²

3-D Dipole −µ

4π

(z − z0)

r³ —

Strength µ

Direction: z-axis − µ 4π

(z − z₀)

{(x − x₀)²+ (y − y₀)²+ (z − z₀)}^3/2 —

Vortex Γ

2πθ −Γ

2πlog_e r

Strength Γ Γ

2πtan⁻¹ y − y₀ x − x0

− Γ

2πlog_ep

(x − x₀)²+ (y − y₀)²

• Typical Velocity Profiles and Streamlines on a Ship Forebody

5.6 Separation of Variables

• Laplace equation is separable in 13 coordinate systems.

(35)

Figure 5.11 Streamlines about a ship forebody. (From Saunders 1957)

(36)

5.6 Separation of Variables 229

– Example): Spherical coordinate (r, θ, α)

x = r cos θ, y = r sin θ cos α, z = r sin θ sin α (5.109) Laplace equation

1 r²

∂

∂r

r²∂φ

∂r

+ 1

r²sin θ

∂

∂θ

sin θ∂φ

∂θ

+ 1

r²sin²θ

∂²φ

∂α² = 0 (5.110)

• Example): Flow around a Sphere in Uniform Stream – Flow is axi-symmetric.

1 r²

∂

∂r

r²∂φ

∂r

+ 1

r²sin θ

∂

∂θ

sin θ∂φ

∂θ

= 0 for r > r₀ (5.111) The boundary conditions

∂φ

∂r = 0 on r = r₀ (5.112)

φ → U r cos θ as r → ∞ (5.113) – Solution is assumed in the form of separation variables:

φ(r, θ) = F (r)G(θ) (5.114)

– The resulting Laplace equation r²φ⁻¹∇²φ = 1

F d

dr r²F⁰ + 1 G sin θ

d

dθ (sin θ G⁰) = 0 (5.115) – Ordinary second order differential equations, with a constant C,

d

dr r²F⁰ − F C = 0 (5.116) d

dθ (sin θ G⁰) + CG sin θ = 0 (5.117) – Solutions of the second equation is Legendre functions.

P_n(cos θ) where C = n(n + 1) (5.118)

(37)

The first three of Legendre functions:

P₀(x) = 1, P₁(x) = x, P₂(x) = 1

2(3x² − 1) (5.119) – The first equation is of the form.

d

dr r²F⁰ − F n(n + 1) = 0 (5.120) where solutions are given by

F = F = rⁿ F = 1

rⁿ⁺¹ (5.121)

– General solution form:

φ =

∞

X

n=0

P_n(cos θ)

A_nrⁿ+ B_n rⁿ⁺¹

(5.122)

– Special case of n = 1 in general form:

φ = U cos θ

r + 1

2 r₀³ r²

(5.123)

– The corresponding general solution form in 2-D:

φ =

∞

X

n=1

(cos nθ)

Anrⁿ+ B_n rⁿ

+ A0 + B0log_er (5.124)

5.6.1 Fixed Bodies and Moving Bodies

• Comparison of Flows for Circle and Sphere of Radius r0: ¹¹

11 Movies

Potential flow for a Fixed Circle: ./mmfm_movies/Circle1.mov

Potential flow for a Moving Circle: ./mmfm_movies/Cylinder_moving.mov

(38)

5.7 Method of Images 231

Table 5.3 Comparison of flows for fixed bodies and moving bodies: circle and sphere of radius r₀

Horizontal Streaming Flow pasta Fixed Body

Circle Sphere

Boundary Condition on Body ∂φ/∂r = 0 ∂φ/∂r = 0

Boundary Condition at Infinity φ → U x = U r cos θ φ → U x = U r cos θ

Velocity Potential φ = U (r + r²₀/r) cos θ φ = U (r +¹₂r₀³/r²) cos θ HorizontalTranslation of Bodywith Fluid at Rest at Infinity

Circle Sphere

Boundary Condition on Body ∂φ/∂r = U cos θ ∂φ/∂r = U cos θ

Boundary Condition at Infinity φ → 0 φ → 0

Velocity Potential φ = − (U r₀²/r) cos θ φ = − ¹₂U r³₀/r² cos θ

5.7 Method of Images

• When a body moves near a plane rigid boundary, such as horizontal wall, the interaction between the body and the wall must be accounted for.

– This is done by imposing the boundary condition ∂φ

∂n on the wall and solving the resulting boundary-value problem for the prescribed motion of the body.

– An elegant approach is to replace the wall by an image body, symmetrically disposed on the opposite side of the wall with a suitably prescribed symmetric motion to ensure that the boundary condition on the wall is satisfied.

(39)

Figure 5.12 Method of images: Source above a wall. (From MIT website 2004)

(40)

• Examples of Image Systems in 2-D

– The flow past a pair of circular cylinders of radius a can be expressed schematically by adding an image of the cylinder.

Figure 5.13 Method of images: Circular cylinder above a wall. (From MIT website 2004)

– The flow situation of a pair of 2-D vortex plus its image in uniform stream.

– The flows about an object inside the L-corner can be represented by the object and 3 images.

– The flow about a body situated between two parallel walls is expressed by image series of the body.

(41)

Figure 5.14 Method of images: Vortex above a wall. (From MIT website 2004)

Figure 5.15 Method of images: Circular cylinder and vortex inside an L-corner. (From MIT website 2004)

(42)

Figure 5.16 Method of images: Circular cylinder in two parallel walls. (From MIT website 2004)

(43)

5.8 Green ’s Theorem and Distributions of Singularities

5.8.1 Scalar Identity

• Apply the divergence theorem for two solutions of Laplace’s equation in a volume V , say φ and ψ: ¹²

I

S

φ∂ψ

∂n − ψ∂φ

∂n

dS = Z

V

∇ · (φ∇ψ − ψ∇φ) dV

= Z

V

φ∇²ψ + ∇φ · ∇ψ − ψ∇²φ − ∇ψ · ∇φ dV

= 0 (5.128)

– For an arbitrary choice of ψ, take the potential at the field point x for a unit source :

ψ = 1

4πr = 1 4π

1

p(x − ξ)² + (y − η)² + (z − ζ)² (5.129)

– For 1

r, the Laplace equation ∇² 1 r

= 0 if r 6= 0.

– ∇²

1

|y − x|

= 0 for a fixed constant vector y, but ∇²

1

|y − x|

= 0 does not exist at x = y. Therefore we exclude this point from the volume.

– Introduce a simply connected region for deriving scalar identity (see

12Green’s first/second identity

Apply the divergence theorem for a vector function u = ψ∇φ Z

V

∇ · u dV = I

S

n · u dS (5.125)

to have

Z

V

ψ∇²φ + ∇ψ · ∇φ dV = I

S

ψ n · ∇φ dS (5.126)

Also apply the divergence theorem for u = φ∇ψ, and combine the two results to have I

S

φ∂ψ

∂n− ψ∂φ

∂n

dS =

Z

V

φ∇²ψ − ψ∇²φ dV (5.127)

(44)

5.8 Green’s Theorem and Distributions of Singularities 237

Appendix for details),

φ(x) = − 1 4π

I

S

φn· (x − y)

|x − y|³ − n· ∇φ

|x − y|

dS (5.130)

• When the field point x is situatedwithin V, a form of Green’s theorem is replaced by

1 4π

I

S+S

φ ∂

∂n

1 r

− 1 r

∂φ

∂n

dS = 0 (5.131)

or 1 4π

I

S

φ ∂

∂n

1 r

− 1 r

∂φ

∂n

dS = − 1 4π

I

S

φ ∂

∂n

1 r

− 1 r

∂φ

∂n

dS (5.132) – The integration region can be interpreted as a single closed surface by

connecting the separate surfaces with a ‘tube’. See Figure5.17 (a).

Figure 5.17 Surfaces of integration for Green’s theorem. (From Newman 1977)

– For sufficiently small , the final limiting value of the integral becomes

− 1

4πφ(x, y, z) I

S

∂

∂n

1 r

dS =⇒ −φ(x, y, z) (5.133)

(45)

– Again if the field point x(x, y, z) is inside V , the field value is φ(x, y, z) = − 1

4π I

S

φ ∂

∂n

1 r

− 1 r

∂φ

∂n

dS (5.134)

• If the field point x is situated on the surface S, the contribution of the integral on a hemisphere S would be a half of the contribution for the whole sphere. See Figure5.17 (b).

φ(x, y, z) = − 1 2π

I

S

φ ∂

∂n

1 r

− 1 r

∂φ

∂n

dS (5.135)

• When the field point x is situatedoutside V, the original form of Green’s theorem is valid because there is no singularity inside the volume V :

0 = 1 4π

I

S

φ ∂

∂n

1 r

− 1 r

∂φ

∂n

dS (5.136)

• Contribution to the integral from the control surface SC far away from a body

– For a body moving in an otherwise infinite and unbounded fluid, the appropriate closed surface S enclosing the fluid volume V must in- clude both the body surface S_B and an additional control surface S_C. – For large spherical radius r, the potential due to the body is of order

r⁻¹, and ∂φ

∂n of order r⁻². The integrand of the integral is of order r⁻³, whereas on S_C the differential area dS will increase in order of r². – The contribution to the integral from SC is of order r⁻¹, and vanishes

in the limit where this surface is an infinite distance from the body.

– For a body moving in an unbounded fluid, the integral can be taken simply over the surface of the body SB.

(46)

5.9 Hydrodynamics Pressure Forces: Forces and Moments Acting on a Body 239

5.9 Hydrodynamics Pressure Forces: Forces and Moments Acting on a Body

• Six components of force and moment vectors F =

Z

SB

p n dS and M = Z

SB

p (r × n) dS (5.137) where the normal vector n points out of the fluid volume and hence into the body.

• Substitute for the dynamic pressure from Bernoulli’s equation to have, F = −ρ

Z

SB

∂φ

∂t + 1

2∇φ · ∇φ

n dS (5.138)

M = −ρ Z

SB

∂φ

∂t + 1

2∇φ · ∇φ

(r × n) dS (5.139)

Figure 5.18 Fixed control surface S_C surrounding the moving body surface S_B. (From Newman 1977)