Unlike the isotropic plate-spring model used in the preceding section the volume-based model allows both vertical and horizontal displacement of the wall to take place in response to the action of the fluid. The wall is assumed to be constructed from an homogeneous isotropic material which extends infinitely in the streamwise direction and is of a finite thickness in the direction normal to the flow. The dynamics of the soft substrate layer are described by the Navier equations and concern the propagation of waves in a viscoelastic material, whilst the thin stiffer outer layer is assumed to be governed by classic thin plate theory. If f is the wall displacement in the ar-direction (parallel to the flow) and rj the wall displacement normal to the flow, then for an isotropic viscoelastic solid the wall motion is described by the following equations.
Ps Ps
s - * ( S * 0 ) -
fd h i d2r)\ U *2 dy2 ) at2 p y at2 G.(K' + 5G‘)
t(2 +
Y y )-(
a
''+
5g
')^(£+S
0 (4.31) (4.32) Here Ga and Ka are the dimensional shear and bulk moduli respectively which are related to the elastic modulus of the material, Es, and the Poisson ratio, */,, as follows.r =
' 2(1 + «/,)’ E ‘ k=
’ 3(1 — 2i/,)E ‘ where ps is the density of the wall body.Viscoelastic damping can be modelled using a complex shear modulus.
G, = C?f(l - ¿7«) (4.33)
where ya is a dimensionless, real damping coefficient (or loss tangent) which can also be modified to incorporate Voigt damping, where ya becomes dependent on frequency.
Expressions for both normal and tangential stresses in the wall are required and for the substrate these are, respectively
<Ts r,
(4.34) (4.35) With the thin, stiff upper layer being modelled using classical thin plate theory, rather than as another viscoelastic layer as in the formulation of Yeo (1988), the corresponding plate
stress terms are d2rj d4rj ~ Ppbdt^ + Bpd ^ .92i “ Ppbw EpbW (4.36) (4.37) where Ep and 6 are the modulus of elasticity of the plate and the plate thickness and Bp is the flexural rigidity of the plate.
B v =” 12(1Epb3 - 1^2)
This approximation introduces a stress discontinuity at the interface of the layers, but as long as the upper plate thickness is small compared to the disturbance wavelength, the thin plate approximation is appropriate. This was validated to some extent in Dixon et al. (1994) where a comparison of free waves is made between a double layer compliant wall with the thin plate approximation and that with the upper layer modelled as another viscoelastic material and revealed little significant difference between the two cases under the small wavenumber constraint.
For purposes of non-dimensionalisation the wall parameters are scaled with respect to fixed quantities; that is, unlike the local displacement thickness, 6”, which is dependent on streamwise location. This ensures that for a given set of non-dimensional values the actual wall parameters remain constant at any point in the streamwise direction. In order to achieve this a fixed reference Reynolds number Rl is introduced:
UoaL
Rl = (4.38)
This provides suitable values of velocity, Uoo, and a length scale, L, which are then used in the non-dimensionalisation process. Uoo is taken to be the value of the freestream velocity at the onset point of the pressure gradient (see Figure 3.1). Then, for example, dimensionless wall parameters are written
G. = G.p ,U l' B." p p,U lLV
» -è
(4.39)where an overbar denotes the non-dimensional value.
On the other hand quantities associated with the flow, such as the flow eigenvalues, are made dimensionless in the usual way via a conventional Reynolds number based on boundary layer displacement thickness.
„ U.S- _ _ w6'
Rt- = --- , v a = ab , u = -r—Ue (4.40)
where Ut is the local freestream velocity in the region where the pressure gradient acts. 90
Disturbances are assumed to be of the travelling-wave form so that the displacement vector can be expressed in a similar manner.
v
= [{(«.it. <).•»(*.». OF = **[£(»)■ v(v)Fe,("-,‘',) + cc-
<4-41)
The displacement vector field can be written as a sum of rotational and irrotational terms.ij = + V A (4.42)
and the Navier equations then reduce to two wave equations for <t> and t/’ ■ d2<l> dt2 d2i/> where c£v2«> (4.43) c£v2v> (4.44) ) p» (4.45)
ci and are respectively the dilational and transverse wave speeds in the material and using the representations
0(x, y, t) = + c.c. VK®. y, i) = [0, V’(y), 0]Te,<'a* “'** + c.c. (4.46) the wave equations reduce further to two ordinary differential equations. The vector form of
is taken to ensure a two-dimensional displacement field consistent with two-dimensional disturbance propagation.
4?' = a2A'£<^ with A-2 - !L ~ cl (4.47)
r = a2Aj.^ with = 1 - J (4.48)
where primes denote differentiation with respect to y.
The solutions to each of these equations are a sum of exponential terms and the non- dimensional amplitudes can then be expressed as follows.
£ = id s m h ( a K 11y)C\ + ici cosh(df\ily)C'2 +
ôKt cosh( & KtIü)C3 + âA'r sinh(âA'7dÿ)C4 (4.49)
rj = dK i cosh(oA'r./y)Ci + ah’i cos\\(aKily)C\ —
iâ sinh(ôA'TÎÿ)C3 — id coah(âKtly)C* (4.50)
The parameter Z is the ratio of the two reference length scales involved in the non- dimensionalisation process which for flow over a flat plate is simply / = L/S* = Ri / Ri• but in this case, owing to the streamwise variation in the freestream velocity, its form is not so simple. In the presence of a pressure gradient
— - —l El. - El (—\ m
S* Ri* Uoo Ri* 'L J (4.51)
where m is the power appearing in the expression for the potential flow, x can be eliminated from Eq.( 4.51) using the expression for 6’ in Eq.( 4.9) to leave
(4.52) In the absence of any pressure gradient m = 0 and this reduces to Rl/R i* as required.
The equations describing the motion of the wall must now be coupled to the equations for the flow in order to determine the correct boundary conditions.