USRE41398EExpiredUtilityPatentIndex 55
Design of viscoelastic coatings to reduce turbulent friction drag
Est. expiryApr 10, 2020(expired)· nominal 20-yr term from priority
B64C 23/00B64C 21/10F15D 1/12Y02T70/10Y02T50/10B63B 1/34
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Claims
Abstract
A method is provided to select appropriate material properties for turbulent friction drag reduction, given a specific body configuration and freestream velocity. The method is based on a mathematical description of the balance of energy at the interface between the viscoelastic surface and the moving fluid, and permits determination of the interaction of turbulent boundary layer fluctuations with a viscoelastic layer by solving two subtasks—i.e., a hydrodynamic problem and an elasticity problem, which are coupled by absorption and compliancy coefficients. Displacement, velocity, and energy transfer boundary conditions on a viscoelastic surface are determined, and a Reynolds stress type turbulence model is modified to account for redistribution of turbulent energy in the near-wall region of the boundary layer. The invention permits drag reduction by a coating with specified density, thickness, and complex shear modulus to be predicted for a given body geometry and freestream velocity. For practical applications, viscoelastic coatings may be combined with additional structure, including underlying wedges to minimize edge effects for coatings of finite length, and surface riblets, for stabilization of longitudinal vortices.
Claims
exact text as granted — not AI-modified1. A method for estimating the reduction in friction drag of a body with a viscoelastic coating as compared to the friction drag of a rigid surface of identical size and shape as said body, during turbulent flow at a specified freestream velocity, said method comprising the following steps, performed in the order indicated:
(I) using boundary conditions for a rigid surface, determining characteristics of a turbulent boundary layer at the specified freestream velocity, said characteristics including the boundary layer thickness, phase speed and frequency corresponding to maximum-energy-carrying disturbances, mean velocity profiles, Reynolds stress distributions, wall shear stress distribution, and friction drag,
(II) selecting a density, complex shear modulus, and thickness of a viscoelastic coating which corresponds to minimum oscillation amplitudes and maximum flux of energy into the viscoelastic coating during excitation by a forcing function substantially identical to the excitation produced by the turbulent boundary layer as determined in step I, and
(III) using oscillation amplitudes and energy flux corresponding to the viscoelastic coating of selected density, complex shear modulus and thickness as determined in step II, determining characteristics of the turbulent boundary layer at the specified freestream velocity, including mean velocity profiles, Reynolds stress distributions, wall shear stress distribution, and friction drag, and
(IV) determining a percentage reduction in friction drag as the ratio of i) the friction drag as determined in step I minus the friction drag as determined in step III, divided by ii) the friction drag as determined in step I.
2. The method of claim 1 , wherein step I thereof includes the following substeps, not necessarily performed in the order indicated:
(a) for a rigid surface of specified geometry, and for a given freestream velocity, U ∞ , solving the equation of continuity:
∂ U i ∂ x i = 0
and the equations of motion for an incompressible fluid and steady flow with constant kinematic viscosity, ν, constant density, ρ, negligible body forces, and gradients of pressure, P:
U j ∂ U i ∂ x j = - 1 ρ ∂ ?? ∂ x i + g i + v ∂ 2 U i ∂ x j ∂ x j
where U i refers to velocity components in the x, y, and z directions (x 1 , x 2 , and x 3 in indicial notation) and where fluid velocities at the surface of the body are zero,
(b) from the velocity field, U i , determined from the solution of the general continuity and motion equations in step 2(a), determining the boundary layer thickness as a function of body geometry, where the edge of the boundary layer is defined as that location where the ratio of mean velocity to the freestream velocity is a constant, β, between 0.95 and 1.0:
U U ∞ y δ = 1 = β
(c) from the solution of the general continuity and motion equations in step 2(a), determining the shear stress, τ w , along the body, the local coefficient of friction drag, and the integral drag, and
(d) estimating the frequency, ω e , and the phase speed, C, corresponding to maximum-energy-carrying disturbances in the boundary layer.
3. The method of claim 2 , wherein in substep 2(b), the boundary layer thickness is approximated as that location where the ratio of the mean velocity to the freestream velocity has a constant value of β= 9975 . velocity has a constant value of β= 0 . 9975 .
4. The method of claim 2 , wherein in substep 2(d), the maximum energy-carrying frequency for disturbances in the boundary layer is estimated as:
ω e = U ∞ δ
and the phase speed corresponding to energy-carrying disturbances in the boundary layer is assumed to be:
C=0.8U ∞ .
5. The method of claim ( 2 ), for the case of two-dimensional flow over a flat plate, wherein the equations of continuity and the equations of motion are reduced to the following system of equations, where a Reynolds transport approach is adopted for turbulence closure, and where P is the mean pressure, ν is the kinematic viscosity, ρ is the density, U is the mean longitudinal velocity component, V is the mean normal velocity component, and u′, v′, and w′ are components of fluctuating velocity in the streamwise, normal, and transverse directions, respectively, and ε is the isotropic dissipation rate:
∂ U ∂ x + ∂ V ∂ y = 0 U ∂ U ∂ x + V ∂ U ∂ y = - 1 ρ ∂ ?? ∂ x i + v ∂ 2 U ∂ y 2 - ∂ u ′ v ′ _ ∂ y U k ∂ u i ′ u j ′ _ ∂ x k = P ij - Π ij - ∂ J ijk ∂ x k - 2 ɛ ij U ∂ ɛ ∂ x + V ∂ ɛ ∂ y = C ɛ 1 f 1 ɛ k P Σ - C ɛ 2 f 2 ɛ k [ ɛ - v ∂ 2 k ∂ y 2 ] + ∂ ∂ y ( 2 ɛ t ∂ ɛ ∂ y ) + v ∂ 2 ɛ ∂ y 2
where:
P ij is the production tensor, expressed as
[ [ P ij = - u i ′ u k ′ _ ∂ U j ∂ x k - u j ′ u k ′ _ ∂ U i ∂ x k P Σ = 1 2 P ii ] ]
P if = - u i ′ u k ′ _ ∂ U j ∂ x k - u j ′ u k ′ _ ∂ U i ∂ x k
P Σ = 1 2 P i i
Π ij is the pressure-strain correlation tensor, expressed as:
Π ij = C 1 ( ɛ k ) ( u i ′ u j ′ _ - δ ij 2 3 k ) + C 2 ( P ij - δ ij 2 3 P Σ ) - ( C i ′ ɛ k ( v ′2 _ δ ij - 3 2 ( v ′ u i ′ _ δ ij ) ) + C 2 ′ ( P ij - 2 3 δ ij P Σ ) - C 2 ′ ( P ij - D ij ) ) R t R k ( 1 + 1 + 100 R t )
where:
k is the turbulent kinetic energy, expressed as
k = 1 2 ( u ′2 _ + v ′2 _ + w ′2 _ )
D ij is the dissipation tensor, expressed as
D ij = - u i u j _ ∂ U i ∂ x j - u j u i _ ∂ U i ∂ x i R t = k 2 v ɛ
is a nondimensional ratio between the square of the kinetic energy and the product of the kinematic viscosity and dissipation velocity
R k = k 1 / 2 y v
is the distance from the wall, nondimensionalized by the ratio of the kinematic viscosity to the square of the kinetic energy C 1 , C 2 , C 1 ′, C 2 ′, and C 3 ′ are constants, defined as C 1 =1.34, C 2 =0.8, C 1 ′=0.36, C 2 ′=0.45, and C 3 ′=0.036
J ijk is the tensor that describes the diffusive flux of the Reynolds stresses, components of which are expressed as:
- ∂ J ijk ∂ x 1 ≈ 0
- ∂ J ijk ∂ x 3 ≈ 0
- ∂ J ijk ∂ x 2 ≈ ∂ ∂ y ( ( A C ) t κ ɛ [ v ′2 _ ∂ u i ′ u j ′ _ ∂ y ] ) ︸ turbulent diffusion + v ∂ 2 u i ′ u j ′ _ ∂ y 2 ︸ viscous diffusion
where A is 6 for v′ 2 , 2 for u′ 2 and w′ 2 , and 4 for − u′v′ , where C 1 is a constant, defined as 0.12
ε ij is the dissipation tensor, expressed as:
ɛ i j = 1 1 + 0.06 R t u i ′ u j ′ 2 k _ ɛ + ( 1 - 1 1 + 0.06 R t ) 1 3 δ i j ɛ
C ε1 and C ε2 are constants, defined as C ε1 =1.45 and C ε2 =1.9
ε t is the coefficient of turbulent diffusion, defined as
ɛ t = C t k ɛ v ′2 _ ,
except in the equation for the isotropic dissipation rate, where
ɛ t = C ɛ k ɛ v ′2 _ ,
and where C ε =0.15,
ƒ 1 =1+0.8e −R t
ƒ 2 =1−0.2e −R t 2 .
6. The method of claim 5 wherein Reynolds stress boundary conditions are derived from values of surface amplitudes, ξ 1 and ξ 2 , calculated from solution of the momentum equation for a viscoelastic material:
u ′2 _ y = 0 = 1 2 ω e 2 ( ξ 1 2 + 2 u * 2 ω e v ξ 1 ξ 2 sin ( φ 2 - φ 1 ) + 1 ω e 2 ( u * 2 v ) 2 ξ 2 2 ) v ′2 _ y = 0 = 1 2 ω e 2 ξ 2 2 w ′2 _ y = 0 = 1 2 ω e 2 ξ 1 2 tan 2 Θ - u ′ v ′ _ y = 0 = 1 2 ω e 2 ξ 2 ξ 1 cos ( φ 2 - φ 1 )
where:
ω e = U ∞ δ
(φ 2 -φ 1 , is the phase difference between normal and longitudinal oscillation amplitudes
Θ is the angle between the mean flow and the x-axis u* is the friction velocity, defined as μ φ is the friction velocity, defined as
u * = τ w ρ ,
where:
ρ is the density of the fluid, and
τ w is the shear stress at the wall, expressed as
τ w = ρ v ∂ U ∂ y y = 0
|ξ i | is the root-mean-square (rms) amplitude of the displacement.
7. The method of claim 6 wherein the energy boundary condition is expressed in terms of the solution of the momentum equation for a viscoelastic material where:
(a) the energy flux into the coating is determined from the pressure velocity correlation, which, given a periodic forcing function can be expressed in terms of the amplitude of the normal oscillation amplitude (and its complex conjugate, as designated by *):
- p ′ v ′ _ y = 0 = - 1 4 p ′ ω ( - ⅈξ 2 + ⅈξ 2 * )
(b) the amplitude of surface oscillations, ξ 2 , given surface loading corresponding to the turbulent boundary layer, is expressed as:
ξ 2 ❘ p rms = actual = ρ K p u * 2 ξ 2 ❘ p rms = 1 H H u ~ * 2 = C k2 H u ~ * 2
where:
H is the thickness of the coating,
u ~ * = u * U ∞ , and
u* is the friction velocity, defined asμ 0 is the friction velocity, defined as u * = τ w ρ ,
where:
ρ is the density of the fluid, and
τ w is the shear stress at the wall, expressed as
τ w = ρ v ∂ U ∂ y y = 0
so that the pressure velocity correlation can be expressed as:
- - p ′ v ′ _ y = 0 ρU ∞ 3 = - 1 4 H δ C k3 u _ * 4
where:
C k3 =C k2 K p γ(ω)
and where K p is the Kraichnan parameter, with a value of 2.5, and γ(ω) is a dissipative function of the coating material, which reflects the phase shift between the pressure fluctuations and the vertical displacement of the coating,
(c) the flux of energy into the coating can be approximated by an effective turbulent diffusion term,
ɛ t ∂ k ∂ y ,
where:
ɛ t = C t k ɛ v ′2 _ ,
where C t =0.12
so that, in terms of nondimensionalized quantities:
- - p ′ v ′ _ y = 0 ρU ∞ 3 ≈ ɛ ~ t ∂ k ~ ∂ y ~ y = 0
where:
ɛ ~ t = ɛ t U ∞ δ k ~ = k U ∞ 2
and:
y ~ = y δ
(d) given the expression for nondimensionalized pressure-velocity correlation in step (b) and in step (c), ε q =ε t | y=0 can be written as:
ɛ ~ q = ɛ t ❘ y = 0 U ∞ δ = 1 4 C k3 ( H δ ) u ~ * u ~ * 3 Δ max k max + - k q +
where γ max + is the normal distance from the surface to the maximum of turbulence energy, nondimensionalized as follows:
y max + = y max v u * Δ max = y + Re *
where:
Re * = u * δ v
is the Reynolds number based on friction velocity, u*, and boundary layer thickness, δ
u* is the friction velocity, defined as
μ 0 , and boundary layer thickness, δ
μ 0 is the friction velocity, defined above as u * = τ w ρ ,
where:
ρ is the density of the fluid, and
τ w is the shear stress at the wall, expressed as
τ w = ρ v ∂ U ∂ y y = 0
and
k max + = k max U ∞ 2
is the maximum of turbulence kinetic energy, non-dimensionalized by U ∞ 2
k q + = k q U ∞ 2
is the kinetic energy of the oscillating surface, non-dimensionalized by U ∞ 2
(e) the boundary condition for isotropic dissipation rate is expressed dimensionally as:
ɛ y = 0 = ∂ ∂ y [ ( v + ɛ t ) ∂ k ∂ y ] y = 0
or, in nondimensional form:
ɛ ~ y = 0 = ∂ ∂ y [ ( 1 Re + ɛ ~ t ) ∂ k ~ ∂ y ~ ] y = 0
where:
Re = U ∞ δ v
is the Reynolds number based on freestream velocity and boundary layer thickness.
8. The method of claim 7 , where the solution of the turbulent boundary layer problem for an isotropic viscoelastic surface is based upon an iterative technique, and where initial values are assumed for u* and for the gradient of turbulent kinetic energy,
∂ k ∂ y y = 0
as based upon the solution of the turbulent boundary layer problem for a rigid flat plate, with two-dimensional flow, where the boundary conditions are:
u ′2 _ U ∞ 2 y = 0 = 0 v ′2 _ U ∞ 2 y = 0 = 0 w ′2 _ U ∞ 2 y = 0 = 0 u ′ v ′ _ U ∞ 2 y = 0 = 0 ɛ y = 0 = v ∂ 2 k ∂ y 2 y = 0
where the values of u* and the gradient of turbulent kineticwhere the values of μ 0 and the gradient of turbulent kinetic energy,
∂ k ∂ y y = 0 ,
obtained from this solution are used to determine boundary conditions for the next iteration, and where this procedure continues until the solution converges.
9. The method for estimating the reduction in friction drag according to claim 8 , for the case of two-dimensional flow over a flat plate coated by an isotropic viscoelastic material, wherein:
a.) the value of the shear modulus, μ(ω), is constant in all directions,
b.) the phase difference between normal and longitudinal displacements at the wall is equal to π/2, so that the Reynolds shear stresses at the surface of the viscoelastic coating are equal to zero:
− u′v′ =0
c) Reynolds stress boundary conditions are expressed in non-dimensional form as:
u ′2 _ U ∞ 2 = 1 2 ( H δ ) 2 ( C k1 + u ~ * Re * C k2 ) 2 u ~ * 4 v ′2 _ U ∞ 2 = 1 2 ( H δ ) 2 C k2 2 u ~ * 4 w ′2 _ U ∞ 2 = 1 2 ( H δ ) 2 C k1 2 u ~ * 4 tan 2 Θ - u ′ v ′ _ U ∞ 2 = 0
where
Re * = u * δ v
is the Reynolds number based on friction velocity, u*, and boundary layer thickness, δ
u* is the friction velocity, defined as
μ 0 , and boundary layer thickness, δ
μ 0 is the friction velocity, defined above as u * = τ w ρ ,
where:
ρ is the density of the fluid, and
τ ω is the shear stress at the wall, expressed as
τ w = ρ v ∂ U ∂ y y = 0 u ~ * = u * U ∞ C ki = ρ K p u * 2 ξ i ❘ p rms = 1 H K p = 2.5
and where C k1 and C k2 can be approximated as zero if the amplitude of surface oscillations is less than the thickness of the viscous sublayer.
10. The method according to claim 8 , wherein the coating is composed of multiple layers of isotropic viscoelastic materials with different material properties, where:
a.) the shear modulus, μ(ω), is constant within each of the multiple layers;
b.) boundaries between layers are fixed, and
c.) the static shear modulus of the material is progressively lower for each layer from the top layer to the bottom layer of the coating.
11. The method according to claim 8 , wherein the coating is composed of an anisotropic material, where the properties of the material in the normal direction (y) differ from those in the transverse (x-z) plane, wherein the shear modulus in the streamwise and transverse directions is given by μ 1 (ω), and the shear modulus in the normal direction is given by μ 2 (ω):
μ 1 ( ω ) = μ 01 + μ s1 [ ( ωτ s1 ) 2 1 + ( ωτ s1 ) 2 + ⅈ ωτ s1 1 + ( ωτ s1 ) 2 ] μ 2 ( ω ) = μ 02 + μ s2 [ ( ωτ s2 ) 2 1 + ( ωτ s2 ) 2 + ⅈ ωτ s2 1 + ( ωτ s2 ) 2 ] .
12. The method of claim 1 , wherein step II thereof includes the following substeps, not necessarily performed in the order indicated:
(a) choosing a density, ρ s , for the coating which is within 10% of the density of water,
(b) selecting an initial choice for the static shear modulus, μ 0 , of the material to avoid resonance conditions, based on the criterion that the speed of shear waves in the material,
μ 0 ρ s ,
is approximately the same as the phase speed, C, of energy-carrying disturbances,
(c) selecting an initial choice for coating thickness, H,
(d) solving the momentum equation for a viscoelastic material:
ρ s ∂ 2 ξ i ∂ t 2 = ∂ σ ij ∂ x j
given harmonic loading of unit amplitude, phase velocity, C, corresponding to the load of the turbulent boundary layer, and variable wavenumber and for a coating which is fixed at its base to a rigid substrate, where:
ξ 1 and ξ 2 are the longitudinal and normal displacements through the thickness of the coating, approximated by the first mode of a Fourier series as follows:
ξ i =a i1 e ia e (x−Ct)
where a i1 is a coefficient, α e =ω e /C is the wavenumber corresponding to maximum energy in the boundary layer, and φ i is a phase difference,
σ ij is the stress tensor, written for a Kelvin-Voigt type viscoelastic material as:
σ ij =λ(ω)ε s δ ij +2μ(ω)ε ij s
where ε ij s is the strain tensor, expressed as:
ɛ ij s = 1 2 ( ∂ ξ i ∂ x j + ∂ ξ j ∂ x i )
where ε s =ε ii s ,
λ(ω) is the frequency-dependent Lame constant, defined in terms of the bulk modulus, K(ω), and the complex shear modulus, λ(ω):
λ ( ω ) = K 0 - 2 3 μ ( ω )
μ(ω) is constant in all directions for an isotropic material, but will have different values in different directions for an anisotropic material,
and where the momentum equation is solved for a series of materials whose shear moduli can be approximated for a single-relaxation type (SRT) of material, characterized mathematically in the form:
μ ( ω ) = μ 0 + μ s [ ( ωτ s ) 2 1 + ( ωτ s ) 2 + ⅈ ωτ s 1 + ( ωτ s ) 2 ]
with an initial value of static shear modulus, μ 0 =μ| ω=0 , determined in step 3(b), and for different values of dynamic shear modulus, μ s , and relaxation time, τ s , where:
μ s =μ| ω=∞ −μ 0
(e) from the series of SRT materials for which calculations were performed in step 3(d), selecting a material where the calculated oscillation amplitude does not exceed the viscous sublayer thickness under given flow conditions, and where the energy flux into the coating is considered over a range of frequencies approximately one decade above and below the energy-carrying frequency, ω e ,
(f) iterating steps 3(d) and 3(e) with different values of thickness and static modulus, and determine the optimal combination of material properties for a single relaxation time (SRT) type of material,
(g) using the results from step 3(f) as a guideline in the frequency range of interest, solving the momentum equation for viscoelastic materials whose complex shear modulus may be characterized in the form of a Havriliak-Negami, or HN, material, whose shear modulus is expressed in the following form:
μ - μ ∞ μ 0 - μ ∞ = 1 [ 1 + ( ⅈωτ HN ) α HN ] β HN
where μ ∞ =μ| ω=∞ , τ HN is a relaxation time, and α HN and β HN are constants, and
(h) selecting final material properties of a coating material based on conditions of minimal oscillation amplitudes and maximum energy flux, which is equivalent to the correlation between pressure and velocity fluctuations, − p′v′ , in the range of frequencies corresponding to maximum-energy-carrying disturbances, as well as based on consideration of material fabricability.
13. The method of claim 1 , wherein step III thereof includes the following substeps, performed in the order indicated:
(a) solving the equations of motion and continuity for a body with a viscoelastic coating, given the same location and flow conditions as in step I, using a numerical methodology which accounts for non-zero energy flux and surface oscillation boundary conditions, as well as for redistribution of energy between fluctuating components in the near-wall region, and
(b) comparing the friction drag calculated for a viscoelastic plate with that calculated for a rigid plate.
14. The method of claim 1 , wherein said viscoelastic coating is configured with multiple surface or internal structures that approximate the natural longitudinal and transverse wavelengths in the nearwall flow.
15. The method of claim ( 1 ), wherein said viscoelastic coating is structured with a rigid underlying wedge or contour shape to minimize coating thickness and oscillations near the intersection of the coating with a rigid surface.
16. A method for reducing the friction drag of a body by providing a viscoelastic coating, said method comprising the following steps, performed in the order indicated:
a ) determining characteristics of a turbulent boundary layer at a specified freestream velocity using boundary conditions for a rigid surface of an identical size and shape of the surface to be coated, said characteristics including viscous sublayer thickness, boundary layer thickness, phase speed and frequency corresponding to maximum-energy-carrying disturbances, mean velocity profiles, Reynolds stress distributions, wall shear stress distribution and friction drag; b ) selecting material properties of the viscoelastic coating including a density, a complex shear modulus and a coating thickness that, when excited by a forcing function substantially identical to that produced by the turbulent boundary layer as determined in step a ) , will provide the maximum flux of energy into the viscoelastic coating without producing surface excitation amplitudes that exceed the viscous sublayer thickness; c) determining characteristics of the turbulent boundary layer over the viscoelastic coating and at the specified freestream velocity using oscillation amplitudes and an energy flux corresponding to the material properties of the viscoelastic coating selected in step b), including mean velocity profiles, Reynolds stress distributions, wall shear stress distribution, and friction drag, d) determining a percentage reduction in friction drag as a ratio of the value of the friction drag without the viscoelastic coating as determined in step a) minus the friction drag with the viscoelastic coating as determined in step c) divided by the friction drag without the viscoelastic coating as determined in step a), and thereby quantifying the performance potential of the coating design; and e) composing the viscoelastic coating from a material or a combination of materials as selected in all of the steps a) through d).
17. The method according to claim 16 , wherein the viscoelastic coating is composed of multiple layers of isotropic, viscoelastic materials with different material properties, where:
a) the complex shear modulus, μ(ω), is constant within each of the multiple layers, b) boundaries between layers are fixed, and c) the static shear modulus μ 0 of the material is progressively lower for each layer from the top layer to the bottom layer of the viscoelastic coating.
18. The method according to claim 16 , wherein the coating is composed of an anisotropic material, and the properties of the anisotropic material in the normal direction (y) differ from those in the transverse (x-z) plane.
19. The method of claim 16 , wherein said viscoelastic coating is structured with a more rigid underlying wedge or a contour shape to decrease the viscoelastic coating thickness and thereby minimize oscillations near the intersection of the viscoelastic coating with a rigid surface.
20. The method of claim 16 , wherein the viscoelastic coating is combined with surface structures to enhance the stabilization of natural longitudinal and transverse wavelengths in the near-wall flow.
21. A viscoelastic coating designed according to the method of claim 16 .
22. A viscoelastic coating designed according to the method of claim 17 .
23. A viscoelastic coating designed according to the method of claim 18 .
24. A viscoelastic coating designed according to the method of claim 19 .
25. A viscoelastic coating designed according to the method of claim 20 .Cited by (0)
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