4A10 Flow instability Fluid-structure interaction CUED IIB Lecture notes 2 Simple models for structures 4 Galloping and flutter

Chapter 3 Simple models for fluids

The fluid flow around objects or through conduits is commonly characterized by a velocity $U$ . This could be the speed of wind or water far upstream, called the freestream speed. Or this could be the average speed of the fluid through enclosed channels, such as pipes and conduits. Without loss of generality, such a speed is a system parameter, which influences the fluid dynamic force on the structure. In addition to the fluid speed, the other relevant properties of the fluid are its density $\rho$ and viscosity $\mu$ .

The sensible reader has immediately realized that the parameters $\rho$ , $\mu$ and $U$ , which characterize the fluid flow, cannot be combined into any dimensionless form by themselves. If the flow is to change character, such as a transition from a stable state to an unstable state, say by itself or when coupled with a structure, then the critical parameter must be dimensionless. The parameter $U$ feels essential (although we may get a chance to examine a situation which is not chacterized by a speed.) Therefore, $U$ must be combined with $\rho$ and/or $\mu$ , along with some more parameters we have not yet recognized. At the very least, the fluid dynamic force on the structure must have the appropriate dimensions and depend on the relevant quanitites in a physically meaningful way. Gaining insight into this dependence is the objective of this chapter.

3.1 Parametrization in the viscous and inviscid limit

The state of the fluid in general is expressed by the Eulerian velocity field $\bm{u}(\bm{x},t)$ . The state evolution is governed by the Navier-Stokes equations (here we will assume incompressible flow, $\bm{\nabla}\cdot\bm{u}=0$ )

\displaystyle\rho\left[{\dfrac{\partial{\bm{u}}}{\partial{t}}}+(\bm{u}\cdot\bm% {\nabla})\bm{u}\right]+\bm{\nabla}p=\bm{f}+\mu\nabla^{2}\bm{u},

(3.1)

where $p$ is the fluid pressure, which enforces the incompressibility, and $\bm{f}$ is the external volumetric force. The term on the left-hand side proportional to $\rho$ represents the inertia of the fluid and the one on the right-hand size proportional to $\mu$ represents the viscosity. A general analytical solution of eq. 3.1 in combination with the required boundary conditions of interest has not been possible and is, therefore, outside the scope of this module. This equation may be solved computationally, and this is a very popular endeavour, one which we will not pursue in this module in the interest of insight.

The rescaling $\bm{u}=U{\tilde{{\bm{u}}}}$ to make $\bm{u}$ dimensionless must necessarily be accompanied by the rescaling of $t$ and $\bm{x}$ , but what would determine the scales for these variables?

Advanced tip.

There is a length scale $\ell=\mu/(\rho U)$ that can be constructed from the fluid parameters. Similarly there is also a time scale $\tau=\mu/(\rho U^{2})$ . However, the relevant question is whether the flow speed varies by the magnitude $U$ over points separated by these length and time scales. It is generally found that these scales are associated with turbulence, which is a purely fluid dynamical phenomenon, once the scale for the flow speed is given some thought and possibly replaced by an apropriate value. Therefore, $\ell$ and $\tau$ do not seem necessarily connected with the interaction of the fluid with the structure.

Generally, the boundary of the fluid and/or the region it occupies introduces scales for length $L$ and time $T$ . These could be the diameter of the sphere or cylinder immersed in the flow, or the chord of an airfoil, or the diameter of a pipe through which the fluid flows. When they are used for non-dimensionalization as $\bm{x}=L{\tilde{{\bm{x}}}}$ and $t=T{\tilde{{t}}}$ , two dimensionless numbers appear in the rescaled eq. 3.1.

\displaystyle\dfrac{L}{UT}{\dfrac{\partial{{\tilde{{\bm{u}}}}}}{\partial{{% \tilde{{t}}}}}}+({\tilde{{\bm{u}}}}\cdot{\tilde{{\bm{\nabla}}}}){\tilde{{\bm{u% }}}}+{\tilde{{\bm{\nabla}}}}{\tilde{{p}}}={\tilde{{\bm{f}}}}+\text{Re}^{-1}{% \tilde{{\nabla}}}^{2}{\tilde{{\bm{u}}}},

(3.2)

where $p=\rho U^{2}{\tilde{{p}}}$ , ${\tilde{{\bm{f}}}}=\bm{f}L/\rho U^{2}$ , and $\text{Re}=\rho UL/\mu$ is the Reynolds number, named after Osborne Reynolds. Of course, if the boundaries move too slowly, so that $T\gg L/U$ , then the response of the flow to that motion may be considered quasisteady and the time-rate-of-change term $\partial{\tilde{{\bm{u}}}}/\partial{\tilde{{t}}}$ may be neglected.

The Reynolds number charcterizes the relative strength of inertia to viscosity in a given situation.

3.1.1 Low Reynolds number flow

When the Reynolds number is small, viscosity dominates over inertia. The inertial term may be neglected to yield the celebrated Stokes equations, named after George Gabriel Stokes, as

\displaystyle-\bm{\nabla}p+\mu\nabla^{2}\bm{u}=\bm{f}.

(3.3)

(In the boundary of the fluid moves sufficiently fast $T\ll\rho L^{2}/\mu$ , then the time-derivative of $\bm{u}$ may not be ignored.) This equation is combined with boundary conditions about the motion of the boundary, where the fluid velocity is specified. The Stokes equations are linear in the velocity and pressure, and, therefore, linear in the relation between the velocity of the boundary and the force on the boundary. In addition, time does not enter the formulation of Stokes flow except through the motion of the boundaries. The flow adjusts instantaneously to the motion of the boundary. (This principle was illustrated by Taylor in 1967[9] in what has now become an iconic video.) The linearity and kinematic reversibility of eq. 3.3 helps immensely in determining the flow and, more importantly, characterizing the forces exerted by such a fluid on strutures. The essence of this argument may be found in the notes by Taylor[8] in this document.

For a general object of length scale $L$ immersed in the fluid, moving with a steady speed $U$ and rotating with angular speed $\Omega$ at low Reynolds numbers, the drag $F_{D}$ , lift $F_{L}$ and torque $Q$ exerted by the fluid may be written as


$\displaystyle F_{D}$	$\displaystyle=C_{t,D}{\mu UL}+C_{r,D}\mu\Omega L^{2},$	(3.4a)
$\displaystyle F_{L}$	$\displaystyle=C_{t,L}{\mu UL}+C_{r,L}\mu\Omega L^{2},\quad\text{and}$	(3.4b)
$\displaystyle Q$	$\displaystyle=C_{t,Q}\mu UL^{2}+C_{r,Q}\mu\Omega L^{3},$	(3.4c)

where $C_{t,D}$ , $C_{r,D}$ , $C_{t,L}$ , $C_{r,L}$ , $C_{t,Q}$ and $C_{r,Q}$ are dimensionless coefficients, which depend on the shape of the body but not its size. For example, for a translating and rotating sphere, $C_{t,D}=3\pi$ , $C_{r,D}=C_{t,L}=C_{r,L}=C_{t,Q}=0$ and $C_{r,Q}=\pi$ , where $L$ is the diameter of the sphere. For details, the reader is referred to the authoritative account by Happel and Brenner[6] on this topic.

3.1.2 High Reynolds number flow

When the Reynolds number is large, inertia dominates over viscosity everywhere in the fluid, except in a thin boundary layer. In this case, the response of the fluid to the presence and motion of the solid structure is much more complicated compared to the case of low Reynolds number. Progress is possible in some cases, such as for streamlined wings based on the application of thin airfoil theory. In other cases, much insight can still be derived from a suitable parameterization of the fluid dynamic forces.

3.1.3 Steady lift on thin airfoils

Two-dimensional potential flow theory applied to thin airfoil explains the flow around real streamlined airfoil cross section for small angles of attack when the flow remains attached. The theory predicts that a thin streamlined cross section of chord $c$ making an angle of attack $\alpha$ with the oncoming flow experiences a lift force $F_{L}$ per unit length perpendicular to the cross section given by

\displaystyle F_{L}=\dfrac{1}{2}\rho U^{2}c~{}C_{L}(\alpha,\text{Re}),\qquad% \text{where}\qquad C_{L}(\alpha,\text{Re})\approx 2\pi\alpha,

(3.5)

where $C_{L}$ is called the lift coefficient. The theory by itself predicts no drag in the inviscid limit. However, when combined with boundary layer theory, the drag per unit length $F_{D}$ is parameterized as

\displaystyle F_{D}=\dfrac{1}{2}\rho U^{2}c~{}C_{D},

(3.6)

where $C_{D}$ is called the drag coefficient. Multiple physical effects contribute to the drag, and an approximate parameterization of $C_{D}(\alpha,\text{Re})$ may be possible in many cases. We will take $C_{D}$ to be given in this module.

This theory only applies in the steady or quasi-steady case, where the time-scale over which the angle of attack changes $T$ is large compared to the aerodynamic time scale $c/U$ , i.e. $UT/c\gg 1$ . When $UT/c\sim O(1)$ or $UT/c\ll 1$ , an unsteady version of the potential flow theory called Wagner’s theory applies. The details of this theory are outside the scope of this module.

3.1.4 Steady bluff body drag

Bluff bodies have shapes for which flows with high Reynolds number do not remain attached to the surface of the body but instead separate. This process of separation causes vorticity to be shed from the boundary layer to the region behind the body and a wake to develop. Due to the separation, the flow deviates significantly from potential flow. The flow in the wake is not strictly steady because turbulence usually develops in this region. (We will discuss the origin of this unsteadiness later in this module when we discuss vortex-induced vibrations.) The flow in the wake may still be considered statistically steady, i.e. steady on average on a time-scale much longer than those of turbulent eddies. Based on this, a parameterization on the average steady drag, $\overline{F}_{D}$ , may be written as

\displaystyle\overline{F}_{D}=\dfrac{1}{2}\rho U^{2}A~{}\overline{C}_{D},

(3.7)

where $A$ is the frontal area presented by the body to the flow and $\overline{C}_{D}$ is the average dimensionless drag coefficient. Equation 3.7 is a parameterization of the drag, which is based on an incomplete understanding of the physics, and, therefore, $\overline{C}_{D}$ is strictly an unknown function of Re, which is measured empirically. An insightful manner of proceeding pretends $\overline{C}_{D}$ to be known, and proceeds to draw conclusions, just to realize that many of the conclusions do not depend sensitively on the precise value of $\overline{C}_{D}$ .

The assumption of statistical steadiness requires a small discussion. On the one hand, the flow changes slowly enough that the parameter $UT/L$ is large, where $L$ is the representative length of the bluff body. On the other hand, the unsteadiness in the wake caused by turbulence is assumed to be fast and statistically stationary so that it may be averaged out. (For phenomena such as vortex-induced vibrations, this type of separation of scales is violated, so watch out for this counterexample.)

In the same spirit as eq. 3.7, an asymmetric bluff body in the flow may also experience a lift force. This lift may be parameterized as

\displaystyle\overline{F}_{L}=\dfrac{1}{2}\rho U^{2}A~{}\overline{C}_{L},

(3.8)

where $\overline{C}_{L}$ is the average dimensionless lift coefficient.

3.2 Simplified versions of Navier-Stokes

In some cases, it is possible to simplify from the three-dimensional nature of Navier-Stokes equations to a one-dimensional. We discuss some such cases in this section.

3.2.1 Force on an unsteady curved fluid-carrying conduit

Figure 3.1: A flexible conduit with centreline following the curve

\bm{X}(s,t)

conveying a fluid with speed

U

uniformly across its cross section.

Consider a flexible conduit, with centreline described by a given unsteady curve $\bm{X}(s,t)$ , where $s$ is the arc-length along the centreline, carrying a fluid with constant density $\rho$ . The shape and area of the conduit cross-section does not vary along its length, and the cross section is assumed to be much smaller than the typical radius of curvature. The fluid is approximated to be inviscid, and to flow with speed $U$ with a uniform profile across the cross section of the conduit. A schematic of such a conduit is shown in fig. 3.1. The objective is to determine the force exerted by the fluid on the conduit.

The fluid velocity relative to the conduit is everywhere tangent to the curve centreline $\hat{\bm{t}}=\partial\bm{X}/\partial s$ , and is therefore given by

\displaystyle\bm{u}={\dfrac{\partial{\bm{X}}}{\partial{t}}}+U{\dfrac{\partial{% \bm{X}}}{\partial{s}}}=\left({\dfrac{\partial{}}{\partial{t}}}+U{\dfrac{% \partial{}}{\partial{s}}}\right)\bm{X}.

(3.9)

The acceleration of the fluid is the material derivative of the velocity

\displaystyle\bm{a}=\left({\dfrac{\partial{}}{\partial{t}}}+U{\dfrac{\partial{% }}{\partial{s}}}\right)\bm{u}=\left({\dfrac{\partial{}}{\partial{t}}}+U{\dfrac% {\partial{}}{\partial{s}}}\right)^{2}\bm{X}.

(3.10)

The force on the conduit per unit length due to the fluid, $\bm{F}$ , is equal and opposite of the force on the fluid, which in turn is equal to the mass per unit length of the fluid times its acceleration as

\displaystyle\bm{F}=-\rho A\bm{a}=-\rho A\left({\dfrac{\partial{}}{\partial{t}% }}+U{\dfrac{\partial{}}{\partial{s}}}\right)^{2}\bm{X}

(3.11)

Question 3.1.

A section of a fluid carrying conduit has a square cross section of side $b$ . In this section, the conduit curves along its length by a quarter circle with a radius of curvature $R$ , which is much longer than $b$ . The fluid has density $\rho$ , and flows with an average speed $U$ through the conduit. The conduit centerline is steady in time. Assuming the fluid to be inviscid and its velocity to be uniform across the conduit cross section, determine the net force it exerts on this section of conduit.

Answer 3.1.

Left as an exercise to the reader. Note that the force may be determined in two ways: (i) Using a momentum balance on a control volume enclosing the fluid going through a quarter turn, (ii) By integrating the force determined using eq. 3.11. Both should yield the same expression for force.

Question 3.2.

A flexible conduit shape oscillates sinusoidally such that its centreline follows the curve given by

\displaystyle\bm{X}=s\hat{\bm{e}}_{x}+a\Re(e^{i\omega t-iks})\hat{\bm{e}}_{y},

(3.12)

where $a\ll 1$ , $k$ and $\omega$ are parameters. The conduit has a circular cross section with cross section area $A$ uniformly along its length. The fluid inside is inviscid, has density $\rho$ and speed $U$ along the centreline. The fluid velocity profile may be assumed to be uniform across the cross section. Determine the force on such a conduit due to the fluid it carries.

Answer 3.2.

Based on eq. 3.11, the force per unit length is

\displaystyle\bm{F}=\rho Aa(\omega+kU)^{2}\Re(e^{i\omega t-iks})\hat{\bm{e}}_{% y}.

(3.13)

3.2.2 Thin layer flow

Consider a two-dimensional case with fluid sandwiched between two walls, as shown in fig. 3.2. The fluid has density $\rho$ and viscosity $\mu$ . The bottom wall is flat and aligned with the $x$ axis but the top wall, possibly made of some solid structure, is curved and unsteady as described by $y=h(x,t)$ . The thickness of the fluid layer is small compared to its scale along the direction of the flow. The objective is to determine the fluid pressure so that the force on the wall at $y=h(x,t)$ may be determined.

Figure 3.2: Fluid flow through a thin layer.

Mass conservation

The fluid is incompressible, so the statement of mass conservation reduces to volume conservation. Using a control volume between $x$ and $x+\textrm{d}x$ yields the relation

\displaystyle{\dfrac{\partial{h}}{\partial{t}}}+{\dfrac{\partial{q}}{\partial{% x}}}=0,\quad\text{where}\quad q=\int_{0}^{h}u(x,y,t)~{}\textrm{d}y.

(3.14)

Here $q(x,t)$ is the flux of fluid volume across $x$ .

When applying momentum balance, we must consider two extreme limits.

Momentum conservation – viscous case

In this case, inertia is negligible, and it is possible to show that by virtue of the thinness of the fluid layer, eq. 3.1 reduces to

\displaystyle{\dfrac{\partial{p}}{\partial{x}}}=\mu{\dfrac{\partial^{2}{u}}{% \partial{y}^{2}}},\quad\Longrightarrow u=-\dfrac{1}{2\mu}{\dfrac{\partial{p}}{% \partial{x}}}~{}y(h-y).

(3.15)

In other words, the film is so thin that locally the flow quickly evolves into the Poiseuille parabolic profile we know and cherish. Here we have assumed a no-slip condition at $y=0$ and $y=h$ , but other boundary conditions may be accommodated without much difficulty. Integrating across the film thickness then yields the flux to be

\displaystyle q=-\dfrac{1}{12\mu}h^{3}{\dfrac{\partial{p}}{\partial{x}}}.

(3.16)

Momentum conservation – inviscid case

If the fluid viscosity is to be neglected, the thickness-wise profile of the fluid velocity must be ascertained ad hoc. A common choice is to assume a top-hat or uniform profile of $u$ across the thickness, so that the volume flux is

\displaystyle q=uh.

(3.17)

Momentum balance in this case invokes the inertia of the fluid in the form

\displaystyle\rho\left({\dfrac{\partial{u}}{\partial{t}}}+u{\dfrac{\partial{u}% }{\partial{x}}}\right)+{\dfrac{\partial{p}}{\partial{x}}}=f-d(u,h),

(3.18)

where $f$ is the external force per unit volume acting on the fluid and $d(u,h)$ is the drag per unit volume exerted by the wall.

The momentum conservation closes the system and provides the fluid pressure in terms of the velocity and the motion of the wall.

Question 3.3.

Onset of roll waves: A uniform thin film of fluid flowing under gravity rapidly down an incline of angle $\theta$ , when perturbed develops undulation in the free surface. Determine the criteria for the onset of such undulations. The wall drag may be taken to be in the form of a so-called Chezy friction coefficient, $C_{f}$ , as

\displaystyle d(u,h)=C_{f}\rho\dfrac{u^{2}}{h}

(3.19)

Answer 3.3.

The question expects us to also develop a model for the flow. We suppose that the word “rapidly” implies a high Reynolds number, and hence are tempted to apply an inviscid thin film approximation. In this case, the top surface of the film is free so the pressure there is atmospheric (taken to be zero). As a result, the pressure at depth $y$ is $p=\rho g(h-y)\cos\theta$ as determined by hydrostatic balance, since the velocity along $y$ is negligible. Substituting this in the inviscid governing equations formed by eqs. 3.14, 3.17 and 3.18 then yields


	$\displaystyle{\dfrac{\partial{h}}{\partial{t}}}+{\dfrac{\partial{(uh)}}{% \partial{x}}}=0,$		(3.20a)
	$\displaystyle{\dfrac{\partial{u}}{\partial{t}}}+u{\dfrac{\partial{u}}{\partial% {x}}}+g\cos\theta{\dfrac{\partial{h}}{\partial{x}}}=g\sin\theta-C_{f}\dfrac{u^% {2}}{h}.$		(3.20b)

What follows is a linear stability analysis of a uniform steady state of these equations.

Note that a uniform steady state of these equations is $u=U$ and $h=H$ such that $g\sin\theta=C_{f}U^{2}/H$ . Perturbing about this steady state as $u=U+u^{\prime}$ and $h=H+h^{\prime}$ , where $u^{\prime}$ and $h^{\prime}$ are assumed to be infinitesimal, yield the following linear equations


	$\displaystyle{\dfrac{\partial{h^{\prime}}}{\partial{t}}}+U{\dfrac{\partial{h^{% \prime}}}{\partial{x}}}+H{\dfrac{\partial{u^{\prime}}}{\partial{x}}}=0,$		(3.21a)
	$\displaystyle{\dfrac{\partial{u^{\prime}}}{\partial{t}}}+U{\dfrac{\partial{u^{% \prime}}}{\partial{x}}}+g\cos\theta{\dfrac{\partial{h^{\prime}}}{\partial{x}}}% =C_{f}\left(\dfrac{2U}{H}u^{\prime}-\dfrac{U^{2}}{H^{2}}h^{\prime}\right).$		(3.21b)

It behooves us to examine perturbations of the form $(u^{\prime},h^{\prime})=(\hat{u},\hat{h})e^{st}e^{ikx}$ , so that the eigenvalue equation for $s$ in terms of $k$ is in the form


	$\displaystyle s\hat{h}+ikU\hat{h}+ikH\hat{u}=0,$		(3.22a)
	$\displaystyle s\hat{u}+ikU\hat{u}+g\cos\theta ik\hat{h}=g\sin\theta\left(2% \dfrac{\hat{u}}{U}-\dfrac{\hat{h}}{H}\right)$		(3.22b)

Now we proceed with the following reasoning. We expect that as $U$ increases, presumably because we are increasing the slope of the incline, at a critical value of $U$ the perturbations will transition from exponential decay to exponential growth. At the transition, the perturbations neither grow nor decay. The complex nature of eq. 3.22 imply that the eigenvalue $s$ must be purely imaginary at the threshold, i.e. say $s=-i\omega$ for some real $\omega$ . With this substitution, eq. 3.22a implies that $(\omega-Uk)\hat{h}=Hk\hat{u}$ , so that $\hat{u}$ and $\hat{h}$ may be taken to be purely real. Decomposing eq. 3.22b into real and imaginary parts then yields $(\omega-Uk)\hat{u}=g\cos\theta k\hat{h}$ and $2H\hat{u}=U\hat{h}$ . Eliminating $\omega$ , $\hat{u}$ and $\hat{h}$ then yields the threshold condition

\displaystyle\dfrac{U^{2}}{gH\cos\theta}=2\quad\text{or, equivalently, }\quad% \tan\theta=2C_{f}.

(3.23)

3.3 Conclusion

To study fluid-structure instabilities, it is necessary to understand and quantitatively represent the force transmitted between the fluid and the structure. It is frequently the case that the motion of the structure modifies the fluid flow, which then reacts by modifying the force exerted on the structure. The process of representing this relationship was considered in this chapter, especially in the form of simplified models. As always, the purpose of these models is to illuminate the underlying physical processes and facilitate insight.