We consider the dynamics of an inextensible two-dimensional elastic plateb of length L, width l, and thickness h ≪ L ≪ l, made of a material of density ρ_s and Young's modulus E embedded in a three- dimensional parallel flow of an ambient fluid with a density ρ_f and kinematic viscosity ν, shown schematically in Fig. 1. We assume that the leading edge of the naturally straight plate is clamped at the origin with its tangent along the x axis and that there are no variations in the flow along the direction of the clamped edge, and that far from the plate, the fluid velocity U = U x. Then the transverse position of the plate Y(x, t) satisfies the equation of motion (10): [1] mY tt=- BYxxxx +lΔP +TYxx Here, and elsewhere A_b ≡ ∂A /∂b, m = ρ_shl is the mass per unit length of the flag, B = Eh³l/12(1 – σ²) is its flexural rigidity (here σ is the Poisson ratio of the material), ΔP is the pressure difference across the plate due to fluid flow, and T is the tension in the flag induced by the flow.

In deriving Eq. 1, we have assumed that the slope of the plate is small so that we can neglect the effect of any geometrical nonlinearities; these become important in determining the detailed evolution of the instability but are not relevant in understanding the onset of flutter. For the case when the leading edge of the flag is clamped and the trailing edge is free, the boundary conditions associated with Eq. 1 are (10): [2] Y (t,0)=0, Yx( t,0) =0, Yxx (t,L)= 0,Y xxx (t,L)=0.

To close the system of Eqs. 1 and 2, we must evaluate the fluid pressure ΔP by solving the equations of motion for the fluid in the presence of the moving plate.

We will assume that the flow is incompressible, inviscid, and irrotational. Then the tension in the flag T = 0,c and we may describe the unsteady fluid flow as a superposition of a noncirculatory flow and a circulatory flow associated with vortex shedding, following the pioneering work of Theodorsen (3). This allows us to respect Kelvin's theorem preserving the total vorticity of the inviscid system (which is always zero) by considering a vortex sheet in the fluid and an image sheet of opposite strength that is in the plate. Both flows may be described by a disturbance velocity potential φ, which itself may be decomposed into a noncirculatory potential, φ_nc, and a circulatory potential, φ_γ, with φ = φ_nc + φ_γ. Then φ satisfies the Laplace equation, ∇ ²φ = 0, characterizing the two-dimensional fluid velocity field, (u, v) = (φ_x, φ_y ), with boundary conditions on the flag, ∇φ·n |_Y=0 = Y_t + UY_x , and in the far-field, ∇φ → 0 as r → ∞.

For small deflections of the plate, the transverse velocity of the fluid, v, varies slowly along the plate. Then we may use a classical result from airfoil theory (11) for an airfoil moving with a velocity v = Y_t + UY_x, assumed to be vary only slightly from a constant, to deduce an approximate form for the noncirculatory velocity potential along the plate as (12) [3] φ nc= x(L- x) [Yt +UY x] This expression neglects terms of order O(UY _xt) (and higher) that correspond physically to the rate of change of the local angle of the plate, which can only be systematically accounted for in a non-local way.d A true check of the validity of our model requires a comparison with the solution of the complete problem, constituting work in progress. However, as we will see, this simple model is able to capture the qualitative essence of the mechanisms involved and agrees reasonably with experiments. Proceeding forward, then, we can use the linearized Bernoulli relation to determine the jump in pressure due to the noncirculatory flow so that [4] P nc=- 2ρf (∂ t φnc+ U∂x φ nc)= ρf U(2x -L) x(L -x) (Y t+ UYx)-2x (L-x)ρ fY tt. Here we note that the fluid added-mass effecte is characterized by the term proportional to Y_tt, and again we have neglected terms of order O(Y_xt) and higher associated with very slow changes in the slope of the plate.

Kelvin's theorem demands that vorticity is conserved in an inviscid flow of given topology. Thus, the circulatory flow associated with vortex shedding from the trailing edge requires a vorticity distribution in the wake of the airfoil and a (bound) vorticity distribution in the airfoil to conserve the total vorticity. If a point vortex shed from the trailing edge of the plate with strength –Γ has a position (L/2)(1 + X₀), X₀ ≥ 1, we must add a point vortex of strength Γ in the interior of the sheet at (L/2)(1 + (1/X₀)). This leads to a circulatory velocity potential along the plate (3–5) φ Γ=- Γ2 πarctan (x (L-x) x 02 -1 L2(1+x 0) -xx0 ), where x ₀ = ((x ₀ + 1/X ₀)/2) characterizes the nondimensional center of vorticity, which is at ((1 + x ₀)/2). Therefore, for a distribution of vortices of strength γ defined by Γ = γ(L/2)dx ₀, the circulatory velocity potential is [5] φ γ= -12 πL 2∫ 1∞ arctan( x(L -x) x0 2-1 L 2( 1+x 0)- xx0 )γ dx0 To calculate the pressure difference due to the circulatory flow, we assume that the shed vorticity moves with the flow velocity U so that ∂_t φ_γ = (2/L)U∂ _x0 φ_γ.f Then, we may write (3) [6] P γ= -ρ fU2 πx( L-x) ∫ 1∞ 2x+L(x 0-1) x 02 -1 γdx0 The vortex sheet strength γ in the previous expression is determined by using the Kutta condition, which enforces the physically reasonable condition that the horizontal component of the velocity does not diverge at the trailing edge:g [7]    ∂x(φ γ+φ nc) |x =L=finite Substituting Eqs. 3 and 5 into Eq. 7 yields the relation [8]   12π ∫1 ∞ x0 +1 x0- 1γ dx0= Yt +UYx Multiplying and dividing Eq. 6 by the two sides of Eq. 8 we obtain [9]    Pγ =-( L(2C- 1)+2 x(1-C )) x(L- x) ρf U(Y t+ UYx), where [10]    C[γ] =∫ 1∞ x0 x 02 -1 γdx0 /∫ 1∞ x 0+1 x0 -1 γdx 0 is the Theodorsen functional (3) quantifying the unsteadiness of the flow. For example, for an airfoil at rest that starts to move suddenly at velocity U, γ = δ(x ₀ – (2/L) Ut) corresponding to the generation of lift due to a vortex that is shed and advected with the fluid. Then [11]    C=1- L2tU We see that as Ut/L → ∞, C → 1, which limit corresponds to the realization of the Kutta condition for steady flow (12). Adding up the contributions to the pressure jump across the plate from the circulatory and noncirculatory flows given by Eqs. 5 and 9 we have ΔP = P_nc + P _γ, i.e., [12]    ΔP=- ρfUC[γ] f(x L) (Yt +UY x)- Lρf n(x L) Ytt , where the dimensionless functions n(s) and f(s) are [13]    f(s) =21 -ss , [14]    n(s) =2( 1-s)s By way of comparison, our consideration of the fluid forces embodied in Eq. 12 differs from that of Crighton and Oswell (13), who treated the case of an infinite plate avoiding many of the complications of a finite plate, and is also different from the treatment of Fitt and Pope ( 8), who neglected the role of fluid added-mass. In the slightly different context of fish swimming, Lighthill (14) considered an undulating slender body such as an eel swimming at high Reynolds number for which he wrote    ΔP=( ∂t +U∂ x)( a(x) v), for the lateral force exerted by the fish on a water slice by equating it to the material derivative of the momentum liberated in the fluid, with a(x) as an ad hoc representation of the apparent mass of the cross section at x, and v = Y_t + UY_x. Our theory differs from this by explicitly accounting for the potential flow field and vortex shedding using the Theodorsen approach in the limit of small deformations when the vertical velocity v is almost constant, and suggests an obvious extension of our work to locomotion.

Substituting Eq. 12 in Eq. 1 (with T = 0) gives us a single equation of motion for the hydrodynamically driven plate [15] mYtt =-BY xxxx - ρfUC[γ] f(xL )( Yt+U Yx) -L ρfn (xL) Ytt , with C[γ] determined by Eq. 10. We note that Eq. 15 accounts for the effects of unsteady flow past a flexible plate of finite length, including vortex shedding and fluid added-mass, but is only valid for small-amplitude motions. To make Eq. 15 dimensionless, we scale all lengths with the length L of the flag, so that x = sL, Y = ηL, and scale time with the bending time L/U_B, where U B=(1/L )( B/m) is the velocity of bending waves of wavelength 2π L. Then Eq. 15 may be written as [16] M ηττ =-η ssss-ρ u0 C[γ]f(s )( ητ+ u0 ηs ) Here M= 1+ρn (s) , where ρ = (lρ_fL/ m) = (ρ_f/ρ_s)( L/h) characterizes the added mass effect and the parameter u ₀ = (U/U_b) is the ratio of the fluid velocity to the bending wave velocity in the plate. We can use symmetry arguments to justify the aerodynamic pressure C[γ] f(s)(η _τ + u₀η_s): the term η_s arises because the moving fluid breaks the s → –s symmetry, while the term η_τ arises because the plate exchanges momentum with the fluid, so that the time reversibility τ → –τ symmetry is also broken. These two leading terms in the pressure, which could have been written down on grounds of symmetry, correspond to a lift force proportional to η _s and a frictional damping proportional to η_τ. By considering the detailed physical mechanisms, we find that the actual form of these terms is more complicated due to the inhomogeneous dimensionless functions f(s), n(s). Thus, understanding the flapping instability reduces to a stability analysis of the trivial solution η = 0 of the system 16, 2 to perturbations as a function of the problem parameters ρ, u₀.

Since the free vortex sheet is advected with the flow, the vorticity distribution may be written as γ = γ((2U/L) (t – t₁) – x₀) and t₁ being the time at which shedding occurs, which in dimensionless terms reads γ = γ(2u₀(τ – τ₁) – x₀). Accounting for the oscillatory nature of the flapping instability with an unknown frequency ω suggests that an equivalent description of the vorticity distribution is given by γ = Re(Ae^{i (ω(τ–τ1)–qxo}), where q = ω/2u₀ is a nondimensional wave number of the vortex sheet. Using the above traveling wave form of the vorticity distribution in Eq. 10 we get an expression for the Theodorsen function (3), [17]    C[γ] =C(q )= H1 (q) H0(q)+ iH1 (q) , where H_i are Hankel functions of ith order. The system 16, 17, 2 completes the formulation of the problem for the stability of a straight flag.