Incompressible Irrotational Flow
Incompressible irrotational flow
Enrique Ortega [email protected] Rotation of a fluid element
As seen in M1_2, the arbitrary motion of a fluid element can be decomposed into
• A translation or displacement due to the velocity. • A deformation (due to extensional and shear strains) mainly related to viscous and compressibility effects. • A rotation (of solid body type) measured through the midpoint of the diagonal of the fluid element. The rate of rotation is defined as the angular velocity. The latter is related to the vorticity of the flow through: d 2 V (1) dt Important: for an incompressible, inviscid flow, the momentum equations show that the vorticity for each fluid element remains constant (see pp. 17 of M1_3).
Note that w is positive in the 2 – Irrotational flow counterclockwise sense Irrotational and rotational flow ij 0 According to the Prandtl’s boundary layer concept, thedomaininatypical(high-Re) aerodynamic problem at low can be divided into outer and inner
flow regions under the following considerations: Extracted from [1]. • In the outer region (away from the body) the flow is considered inviscid and irrotational (viscous contributions vanish in the momentum equations and =0 due to farfield vorticity conservation). • In the inner region the viscous effects are confined to a very thin layer close to the body (vorticity is created at the boundary layer by viscous stresses) and a thin wake extending downstream (vorticity must be convected with the flow). Under these hypotheses, it is assumed that the disturbance of the outer flow, caused by the body and the thin boundary layer around it, is about the same caused by the body alone. Therefore, the inner viscous region is neglected and the entire flow field is assumed inviscid and irrotational. This flow is called ideal flow and its solution is relatively simple and fairly accurate for most aerodynamic problems.
3 – Irrotational flow Some topological notions • Reducible curves: can be contracted to a point without leaving the region . • Simply connected region: a region where any closed curve is reducible. • Barrier: is a curve inserted into the region, but which is NOT part of the resulting modified region. The insertion of barriers can make a region simply connected. • Degree of connectivity: the number of barriers needed to make the region simply connected + 1.
reducible
barrier to
not reducible is now simply connected (degree 2). 4 – Irrotational flow Vorticity and circulation Considering a curve C enclosing a simply connected surface S in the flow field, by Stokes’ theorem it is possible to state Vn dS w n dS SScirculation V ds (2) C This can be illustrated by assuming a simple flow contained in the xy-plane
vu V duxvxs ()() y u y xv y C xy vu xy wdS vorticity = circulation / area S z xy
Note: by mathematical convention, the line integrals are positive in the counterclockwise direction. However, in aerodynamics it is convenient to consider positive in
Extracted from [2]. the clockwise sense.
5 – Irrotational flow Kelvin’s theorem: conservation of the circulation Kelvin’s theorem states that the circulation along a closed fluid curve in an inviscid, barotropic flow with conservative body forces remains constant with time, i.e.
DC D V ds 0 (3) Dt Dt C Note that Eq. (3) is similar to say that an irrotational flow remains irrotational if: • There is not net viscous forces along the path described by the fluid curve (this would cause diffusion of vorticity and, thus, changes in the circulation). • Conservative body forces (e.g. gravity) act through the center of mass of the fluid particle (there is no rotation; thus vorticity is not affected). • The flow is barotropic ( = const. or is a function of p only isentropic flow). For example, if depends on T, its gradient may cause rotation.
6 – Irrotational flow Generation of circulation
Extracted from [3]. Kelvin’s theorem does not impede that circulation can exist in the case of initially the latter is zero. Supposing an airfoil suddenly started
from rest, it is necessary for airfoil0 the existence of an additional circulation, generated at the starting moment (starting vortex), which makes that the total circulation does not change, i.e. it remains zero.
In Figure (b), the curve C2 is split into two loops, one enclosing the starting vortex and the other enclosing the airfoil. Circulation exists along these loops, but the total circulation remains zero; thus, Kelvin’s theorem is satisfied.
7 – Irrotational flow Irrotational flow and the velocity potential Considering the irrotational flow given in the figure, from Eq. (2) we have
2 V ds 0 C
xx21 x 2 x 1 VV ddss V d s V d s(3) xx x x 12 1 2 1 Since the integration paths are arbitrary, to satisfy Eq. (3) it is necessary that the integral is a function only of the integration limits, i.e.
x2 V dsx() () x(4) x 21 1 Therefore, the integrand must be an exact differential of some function of the spatial coordinates. Typically we define V dds (5) where the function (x,y,z) is called the velocity potential.
8 – Irrotational flow Expanding Eq. (5) we have V dudxvds y wdz d dx dy dz (6) x y z Since these expressions are identical, it follows that uvw;; (7) x y z which in vector notation can be expressed by V = . In cylindrical and spherical coordinates, Eqs. (7) result 1 (8) vvrz;; v rr z
11 (9) vvr ;; v rr rsin
9 – Irrotational flow Note that the potential function has dimension L2/T. Potential flow
As seen in M1_3, for a steady, inviscid, incompressible irrotational flow the fluid governing equations can be reduced to V 0 (10)
(0V (11)
Introducing the velocity potential, V = and thus x 0, which satisfies Eq. (11). Then, the continuity equation reduces to the potential flow equation 2 0 (12)
which must be solved along with suitable boundary conditions according to the problem under issue. Once the potential function is determined, the component of the velocity can be obtained from Eqs. (7) and the pressure can be computed using the Bernoulli’s equation.
10 – Irrotational flow Boundary conditions can be of Dirichlet (), Neumann (/n) or mixed type. Some typical boundary conditions used for aerodynamic problems are:
• Infinity boundary conditions: far away from the body the flow approaches the uniform freestream conditions (13) x : U
If the flow is aligned with the x direction: =U∞x (note that the freestream potential is not uniquely determined, as an arbitrary constant can be added it can be assumed to be zero).
• Wall boundary conditions: the flow cannot penetrate the body surface (airtight). This condition translates into U nˆ nˆ 0 (14) along the body surface.
11 – Irrotational flow Remarks • Eq. (12) is a Laplace’s equation, which is a second order PDE of elliptic type (boundary value problem). • The solution is smooth, but singular points are allowed in the domain. • As the equation is linear, the superposition principle is valid, i.e. any linear combination of solutions is also a solution of the problem. • The use of superposition principles allows constructing complex flow patterns by adding elementary flow functions such as uniform rectilinear flow, sources and vortices. • As the potential flow equation does not depend on time, the velocity field adapts instantaneously to changes in the boundary conditions (this means that the velocity of the sound is infinite). No initial conditions are required. • Laplace’s type equations can be also found in other fields such as heat conduction, magnetism, elasticity, electricity, etc. Thus, solutions in one field are generally used in other fields.
12 – Irrotational flow Simply and double connected regions For a simply connected region (e.g. not enclosing an airfoil) in potential flow the Stokes’ theorem says that (see Eq. (2))
V dddss 0 (15) CC C and thus is single-valued. Now, suppose a barrier is inserted to make the region enclosing an airfoil simply connected. The same result should apply to , hence
BA
V dddds VVV sss 0 C1 CCAB 12 A B does not depend on CC12 the path but can be 0! C2
If C1 and C2 are 0, Eq. (15) is not valid along the curves and the potential is not single-valued. Thus, different valid solutions can exist for different values of (it must be specified as a part of the problem).
13 – Irrotational flow Stream function
Considering two arbitrary streamlines in a steady flow, the velocity along these lines must satisfy
V dl udz wdx 0 (16)
and hence the flux between the lines is constant. This can be expressed by BB Vndl udz wdx (17) AA Now, suppose a scalar function (x,z) is defined such that uw; (18) z x and substituting Eqs. (18) into Eq (16), along a streamline we obtain dz dx d 0 (19) = const. zx
14 – Irrotational flow In addition, between two different streamlines d is equal to the volumetric flow rate (see Eq (17)). Introducing Eqs. (18) into the incompressible continuity equation it is possible to verify that uw V 0 (20) xzxzzx
Then, assuming that the flow is irrotational wu 2 (21) wy 0 0 xzx x zz Remarks: • The lines = const. are streamlines of the flow and /n=V. • Compressible and incompressible forms can be derived; stream functions are also valid for viscous flows. • Suitable boundary conditions must be defined.
15 – Irrotational flow Typical boundary conditions for aerodynamic problems in terms of the stream function can be:
• Infinity boundary conditions: for example considering the flow aligned in the positive x direction xu:and0 U w (22) zx
• Wall boundary conditions: for inviscid flows the body contour is a streamline of the flow, thus is constant along the body. Alternatively, it can be enforced that
U nˆ 0 (23) s where s is a distance measured along the body surface.
16 – Irrotational flow Relationship between velocity potential and stream function Assuming the flow to be steady and irrotational, comparing Eqs. (7) and (18) it is possible to obtain the so-called Cauchy-Riemann conditions uw; (24) xz z x which relate the potential and stream functions. Lines of constant potential and streamlines can be shown to be orthogonal; for example it is easy to prove that 0 (25) xx zz Remarks: • Both velocity potential and stream function can be used to model irrotational flows (note that stream function is also valid for rotational flows). • Stream functions are generally restricted to two-dimensional flows (more than one function should be required to describe 3D cases).
17 – Irrotational flow In cylindrical coordinates (in the plane r-), the components of the velocities can be written in terms of the stream function as follows 1 vv; (26) r rr Then, the Cauchy-Riemann conditions (24) result
1 uv; r xz rr (27) 1 wv; zx r r
18 – Irrotational flow References
1. Karamcheti K. Principles of ideal-fluid aerodynamics. R. E. Krieger Pub. Co (1980). 2. Katz J., Plotkin A. Low speed aerodynamics: from wing theory to panel methods. McGraw-Hill series in aeronautical and aerospace engineering (1991). 3. Anderson J. D. Jr. Fundamentals of aerodynamics. McGraw-Hill Book Company (1984).
19 – References and complementary material Enrique Ortega [email protected]