A note on the Galilean invariance of aerodynamic force theories in unsteady incompressible flows

An-Kang Gao·Jiezhi Wu

Abstract As a basic principle in classical mechanics, the Galilean invariance states that the force is the same in all inertial frames of reference.But this principle has not been properly addressed by most unsteady aerodynamic force theories,if the partial force contributed by a local flow structure is to be evaluated.In this note,we discuss the Galilean-invariance conditions of the partial force for several typical theories and numerically test what would happen if these conditions do not hold.

Keywords Galilean invariance·Aerodynamic force theory·Unsteady flow·Flow diagnoses

1 Introduction

Flapping is a common manner in animal locomotion at low Reynolds numbers.By flapping their wings,insects can produce much larger force than steady wings[1–3].This enables species such as bumblebee,which is proclaimed to be unfit to fly by steady aerodynamics,to stay aloft[4].

The high-lift mechanisms in unsteady flows have been explored extensively[3,5–8].It has been widely recognized that vortices shed from the wing are vital for the high lift.Li and Lu [9] further found that vortical structures close to the body dominate force generation. However, due to the high unsteadiness of the flow and complexity of vortex motions[10,11],a quantitative evaluation of the partial force contributed by a specific flow structure is not an easy task,and many theories have been developed for this aim [12].One common feature of these aerodynamic force theories is that the wall-stress integral in calculating the total force is replaced by field integrals, because the flow field contains much richer information than the wall surface does. In so doing, a direct link can be established between flow structures and aerodynamic force.

In classical mechanics,the Galilean invariance is a basic principle that states the force is independent of the choice of inertial frame of reference. The total force is certainly Galilean invariant. Unfortunately, the partial force may not be, since most of the integrands of the field-integral force theories are not Galilean invariant. This means that, when judging the partial force contributed by a local flow structure,the result may be nonunique if different frames of reference are used, which weakens the objectivity of the result and has become a pressing problem that hinders the applications of the theories. This situation motivates us to explore the sufficient and necessary conditions for ensuring the Galilean invariance of the partial force, and to examine what would happen if these conditions do not hold.

Since each theory has its own view in analyses,the comparisons of different theories are not our concern.In addition,our discussion is limited to unsteady flows,because in steady flows there exists a special frame of reference where the flow is steady and the force theories can be greatly simplified.

2 Definition for the Galilean invariance of partial force

Consider a body B submerged in an incompressible flow with constant density.The flow velocity is u,the vorticity is ω = ?×u, and the viscous coefficient is μ. The material derivative D/D t will be denoted by D. Let ?n be the unit normal vector pointing out of the body,the force exerted on the body can be calculated by wall-stress integral as

This formula is convenient in computational fluid dynamics,but it does not reveal which flow structure is important to the force.Therefore,to link the aerodynamic force with flow structures,the control-volume integral is introduced.

As shown in Fig. 1, the control volume Vfis bounded internally by ?B and externally by,with n being the unit normal vector pointing out of the Vf.On the body surface?B,there is n =If the V f contains the whole fluid region,i.e.,is infinitely far from the body,the V f is also denoted by V∞.For simplicity,we assume the Vfis a material fluid volume,so Vfis Galilean invariant itself.

In general, a field-integral force theory is constituted by a body-surface integral, an outer-boundary integral, and a volume integral

The integrand Ivol= Ivol(x,u,ρ,p,μ, D u,?u) represents the force contributed by local flow structures. Let Ω be a material subregion in Vfand ?Ω its boundary,then,the partial force contributed by flow structures in Ω is

In unsteady flows,there is no optimal frame of reference.To describe the fluid motion,assume we choose two inertial frames of reference with coordinates x and ~x respectively.The Galilean transformation between them is

and the velocities are transformed by

Note that ρ, p, μ, D u, and ?u are all Galilean invariant.For simplicity, the dependence of Ivolon these Galileaninvariant variables is omitted, so the simplified notation Ivol= Ivol(x,u)is used henceforth.

The Galilean relativity asserts that the force is the same viewed in all inertial frames of reference.Hence,there should be

for any constants x0and U0.Equation(6)does not unconditionally hold for most force theories,and the sufficient and necessary conditions to ensure Eq. (6) will be given in the next section.There,the following two lemmas will be useful.

Lemma 1Suppose f(u)is a linear function of u and that S is a line segment,surface,or volume.Then,f(u)d S holds for any constant U0if and only if

where ?udenotes the gradient with respect to u.

Lemma 2Suppose f(u) is a linear function of u and that S is a line segment,surface,or volume.Let°represent the dot product,cross product,or tensor product of two vectors.Then,f(u+U0)°(x+x0)d S =f(u)°x d S holds for any constants x0and U0if and only if the following three conditions are satisfied simultaneously

Here,?udenotes the gradient with respect to u.

These two lemmas can be proved using f(u + U0) =f(u)+U0·?uf(u)and f(u+U0)°(x+x0)= f(u)°x+ f(u)°x0+U0·?uf(u)°x+U0·?uf(u)°x0,and then let the integrals of these coefficients vanish.

Strictly speaking, the surface integral over ?B andin Eq.(2)should also be Galilean invariant[13],but this property does not unconditionally hold for most theories either.To derive the Galilean-invariance conditions for the surface integrals, the same method used in dealing with the volume integral can be adopted. For clarity, we only discuss the volume-integral term in this note.

3 Galilean-invariance conditions for several typical aerodynamic force theories

The sufficient and necessary conditions to ensure the Galilean invariance of FΩare specified in this section for several typical aerodynamic force theories.For convenience,we number the theories(Ts)by n(n =1,2,...)in logical order,and add a subscript n to FΩand Ivolfor the n-th theory.

3.1 Force-element theory

In the force-element theory[14–16],the V f is V∞.A singlevalued auxiliary function φ j is introduced. The φ j satisfies

in the flow field.On the body surface,n·?φj=-ej·n;at infinitely far field,φ j goes to zero.Here,(e1,e2,e3)are the unit basis vectors of the coordinate system.The force in the j-th direction is

with l =ω×u being the Lamb vector and

It can be proved that

Hence using Lemma 1,the sufficient and necessary condition for the Galilean invariance of FΩ,1·e j is

3.2 Weighted pressure-source theory

In the weighted pressure-source theory[17],the force in the j-th direction is

with I?B,2=ρφ j D u·n-μφjn·?2u+μ(n×ω)·e j and Q =-0.5?u:?u.Note that Q ＞0 is also used to identify vortical structures,known as the Q-criterion.Although Q is less compact than vorticity, the integrand Ivol,2is Galilean invariant unconditionally,so is FΩ,2.

3.3 Unsteady vortex-force theory

In the unsteady vortex-force theory[18],the force is

with k+1 being the space dimension. FBand Fare

and

respectively.

Using the derivative momentum transformations [18], it can be proved that

with L =ρl+μ?×ω.Since ?u(n×L)=ρ(nω-eiein·ω), the sufficient and necessary condition for the Galilean invariance of FΩ,3can be derived using Lemma 2 as

3.4 Kinetic force theory

In the kinetic force theory[18],the force is

It can be proved that

Using Lemma 2, the sufficient and necessary condition for the Galilean invariance of FΩ,4is

3.5 Impulse theory

The impulse theory is developed independently by Burgers[19],Wu[20],and Lighthill[21],and it has been widely used in unsteady aerodynamics[3,9,22].In the impulse theory,the force is

The sufficient and necessary condition for the Galilean invariance of FΩ,5is

i.e.,the total vorticity in Ω is zero.

In Ref.[23],a minimum-domain impulse theory is developed to minimize the integral domain by excluding all the discrete vortical structures from V∞.The force is

In two-dimensional (2-D) flows, F,6= F,4. Generally,it can be proved that

Since ?u(n× L -ρωn·u) = -ρeiein·ω,the sufficient and necessary condition for the Galilean invariance of F,6is

which reduced to Eq.(21)in 2-D flows due to n·ω=0.

4 Numerical results

The abruptly accelerated flow past a 1/8 ellipse at 45°angle of attack is simulated using the spectral/hp element method[24],with Reynolds number R e=ρU0c/μ=1500,see Fig. 1. ρ = 1, the incoming flow velocity is U0= 1,and the chord length is c = 1. For details and validations of the numerical method see Refs. [17,24]. The lift coefficient and vorticity field at t = 0.3 are shown in Fig. 2a, b,respectively.

Fig. 2 a Lift coefficient of the ellipse. C L p is the pressure force.b Vorticity field at t =0.3.Dashed line is negative

Fig.3 Partial forces contributed by regions G and H, viewed in two frames of reference

Two frames of reference are used.As shown in Fig.1, x is the coordinate in the frame moving with the body,and ~x is that in the frame moving with the incoming flow.There are x =and u=.At time t =0.3,a circle G ={(x,y)|x2+y2≤1 and 65x2+65 y2+126x y ≥0.5}(see Fig.1)and a half circle H ={(x,y)|(x,y)∈G and xy ≤0}(see Fig.2b)are used as the material control volumes.

The circle G contains all the vorticity with zero circulation and more than 99.5% of Ivol,2.Numerical results show that G satisfies all the Galilean-invariance conditions at an accuracy of 10-4, which equals zero approximately. Therefore,FG,n(n = 1,2,...,6) are all Galilean invariant, which is confirmed by numerical results of the lift coefficient(scaled bysee circle symbols in Fig. 3. The relatively large gap between FG,1(or FG,2)and CLis caused by the viscous term in the wall-surface integral,which in this case is of O(R e-0.5)～0.03.

The half circle H contains the leading-edge vortex, but its total vorticity is nonzero and varies with time.So, FH,n(n ／= 2) are not Galilean invariant and hence not uniquely defined, see left-triangle symbols in Fig. 3. This situation should be avoided in applications.

5 Conclusion

The Galilean invariance of the aerodynamic force theories in unsteady incompressible flow is discussed.Although the total force is Galilean invariant,it is proved that the partial force contributed by a local flow structure may not be Galilean invariant for most theories. The Galilean-invariance conditions for several typical theories are found and numerically checked.

AcknowledgementsThis work was supported by the National Natural Science Foundation of China(Grant 11472016).