A rotating three-dimensional body in coupled fluid motion near a wall: Stabilization and blow up

In a coupled fluid–body interaction, the motion of both a freely moving body and the fluid within which it is situated are considered to interact and mutually affect each other. Below, we investigate an extension to recent three-dimensional work by Smith and Liu,¹ which considers a free, finite body located above a solid wall in an incident unidirectional fluid flow over the wall. An unsteady three-dimensional interaction occurs because the fluid flows between the wall and body and over the body in spanwise and streamwise directions. Here, we introduce the effect of body rotation, akin to the flight of a Frisbee.

The real-world motivation for studying such fluid–body interactions arises from industrial, biomedical, and environmental interests, such as aircraft icing,^2–6 or environmental transport such as pollen motion^7,8 and agricultural spray chemical application.^9,10 Typically, such problems are studied through two-dimensional modeling and through a variety of settings and methods, such as direct numerical simulations in near wall flow,^11,12 pipe/channel flow scenarios,^13,14 dispersed particle/particulate flows,^15–20 experimental endeavors,^21,22 and in skipping and skimming interactions.^23–28 

Within two spatial dimensions, several recent studies have furthered this area of research for high Reynolds numbers (as the scales of background motivations often dictate), through modeling and analysis for inviscid fluid–body interactions in boundary layers and channel flows.^11,29–31 Overall, few studies to date have been performed or presented in three spatial dimensions, other than,¹ despite the importance and relevance of three-dimensional effects in the practical motivations above. This sets the scope and necessity of the present study, which is focused on an applied mathematical investigation of a dynamic fluid–body interaction in three spatial dimensions.

Section II below describes the formulation of the model, including the scales, derivation of the unsteady three-dimensional system, and inclusion of body rotation. The model is then simplified with a linearized approach3 for an initial assessment of the parameter space and role of body rotation. Section III discusses the extension of the numerical method by Smith and Liu,¹ which is applied here, while analytical and numerical cases are presented in Sec. IV with consideration of high-frequency body rotation. Section V provides further discussion and conclusions.

The freely moving three-dimensional body is submerged within a laminar, uniform, inviscid, incompressible fluid flow near a wall (or ground). The thickness of the underbody shape is here taken to be the same order as the thickness of the fluid-filled gap between the wall and the underbody but much less than the representative body lengths in the streamwise and spanwise directions. Our coordinate system moves horizontally with the body, as such the wall is vertically non-stationary in this frame of reference. Figure 1 provides a schematic of the scaled setup and indicates the main new features of the system modeled here, namely prescribed body rotation about the Z-axis and surface roughness, e.g., a small surface mounted mound.

Below, the non-linear formulation for the fluid–body interaction in three dimensions is described, wherein the tilde sign denotes dimensional quantities. The body is assumed to be thin, lying close to a fixed solid wall given by $z̃=0$ as in Fig. 1, and the slopes of the body surfaces are considered to be suitably small. The surrounding fluid is regarded as inviscid and incompressible (the implications of which are discussed in the conclusions). Its oncoming flow relative to the body comprises a unidirectional uniform stream $ũ=ũ∞$ in the $x̃$ direction, where $(x̃,ỹ,z̃)$ are Cartesian coordinates with corresponding fluid velocity components being $(ũ,ṽ,w̃)$⁠. Time is denoted by $t̃$⁠, the pressure is denoted by $p̃$ and is taken to be zero in the far field without loss of generality. Finally, $ρ̃F$ and $ρ̃B$ are the constant densities of the fluid and solid body, respectively.

In summary, (5a) ensures that the flow remains tangential at the solid wall, (5b) provides the kinematic condition at the moving underbody surface F for relatively slow scaled time T, and (5c) matches the oncoming flow in the far field. Thus, the fluid flow remains quasi-steady in the gap, in contrast with the motion of the body which, as described in (11) below, is fully unsteady with scaled time variable T of order unity.

As for the remaining conditions, at the leading edge of the planform, (5d) arises from a thin Euler-like region along the body's leading edge where $σ$ is the angle between the tangent to the curved leading edge and the outer stream direction (X-axis), such that $cot σ=X1′(Y)$⁠.³² Finally, constraint (5e) associates with a Kutta-like requirement that the pressure remains continuous across the trailing edge of the planform^30,32 and that P is zero for any trailing edge location at the boundary of the planform. This is because the body's thinness induces only small variations in P except in the gap where it is of order unity. Further discussion of these two conditions is provided in Ref. 1.

Specifically, the body moves freely according to forces and moments acting on the underbody, which is a result of the fluid pressure upon the underbody. Hence, under the above scales, the only relevant forces/moments are given by (i) the scaled lift force $LZ$ acting in the normal direction, (ii) the scaled moment $MY$ acting about the Y-axis, and (iii) the scaled moment $MX$ about the X-axis, with all other forces/torques negligible due to magnitude and timescale. Following from Newton's second law, the body force and moments equate to the area integral of the pressure over the underbody as follows:

For the expansion of pressure in (13b), the leading order term of zero follows from (5c)–(5e) and the $ε$-order term arises due to the comparison of $H,Θ1,Θ2$ each with P in (11). For the expansion of $ϕ$ in (13a), the leading order term here follows from the boundary condition (5c), with the $ε$-order term arising due to the comparison of P and $Φ$ in (4a) and (4b).

Substitution into (8), (9a), (9b), and (10) provides the following system to leading order for the perturbation potential $ϕ(x,y,t)$⁠:

Finally, within this linearized setting, the underbody shape perturbation of (3) and body motion (11) become

The governing equations and boundary conditions of the interaction are now thus (14a–15); initial conditions on $h,θ1,θ2$ and their temporal derivatives $h′,θ1′,θ2′$ are assumed known at time zero. For prescribed underbody $g(x,y,t)$ and planform shape, undergoing a constant rotation about the z-axis, $θ3(t)=ωt$⁠, the above constitutes a closed problem for $ϕ(x,y,t)$⁠, with the locations of the inflow and outflow edges (leading and trailing edges) taken as known in advance for the current linearized setting.

Two observations are particularly worth mentioning: (i) The body's roll angle $θ2$ in our linearized model has no influence on the fluid flow and pressure due to the dominance of the horizontal scale as seen in (14a), (15a), and (ii) the body's vertical displacement $h(t)$ in (15a) does not affect the pitch angle $θ1(t)$ for the same reason. Of interest now are the effects of body rotation about the z-axis, and how such motion for a non-radially symmetric underwater shape can influence the coupled fluid–body interactions. Added complexity arises in the numerical scheme since the leading and trailing edges of a non-circular body planform also vary in time.

Having formulated the equations of motion for the fluid–body interaction, (14) and (15), we now present a method for computationally solving them. In this instance, a finite difference approach will be discussed that uses a fixed-mesh framework to reduce the computational complexity.

We present an extension of both the problem and the methodology formulated in Smith and Liu¹ to include rotation about the z-axis of a body with an elliptical projected planform. The fluid flow in the gap between the body and the wall is of interest, especially for a non-smooth, non-rotationally symmetric underbody surface function (15a). If the body were radially symmetric about the z-axis, the rotation of the body would not affect the fluid–body motion significantly since the x-derivative of f would remain constant in time as the body rotates.

The problem to solve is the Poisson Eq. (14a) with mixed boundary conditions (14b), (14c) prescribed on the evolving leading and trailing edges, $x1(y,t)$ and $x2(y,t)$⁠, of a rotating body. At each time step, the finite difference discretization of this system is encoded as the coefficients of a sparse linear equation matrix, see Smith and Liu¹ for the discretization scheme. However, the rotating body introduces programming complexity.

In a frame of reference with a fixed flow direction, the sparse finite difference matrix formulated for this problem needs to be updated at every time step due to the body rotating. This process can become computationally inefficient, depending on the implementation, since it requires a rewriting of a sparse matrix at each time step. As the size of the finite difference matrix increases (e.g., smaller spatial discretization), the computational costs become significant.

Consider a frame of reference fixed on the body such that the flow direction rotates relative to the body. In this instance, a finite difference matrix that is fixed over time (discussed below) can be derived, thereby circumventing the evolving flow domain challenge. To proceed in this fixed body setting, the (i) coordinate system of the flow and (ii) relative spatial derivatives need careful consideration, as does (iii) the application of boundary conditions at the body leading and trailing edges (⁠ $x1(t)$ and $x2(t)$⁠).

Therefore, the derivatives with respect to x in the equations of motion (14a), (14c), (14d) take the body rotation into account.

Furthermore, given the fixed-body frame of reference, the boundary conditions now rotate about the body boundary where previously they had remained fixed in time for the non-rotating case. Two conditions are applied to the body at the trailing and leading edges, $x1(t)$ and $x2(t)$⁠, respectively. These two edges meet at the two points on the boundary where $dy/dx=0$⁠, i.e., at the body tangents that align with the flow direction. For the computational application of the boundary conditions, we only require the polar coordinate of these interface points, denoted $β1,β2∈[0,2π)$⁠. At the trailing edge, a Dirichlet boundary condition (14b) is applied when $θ∈(β1,β2)$⁠. In contrast, at the leading edge, a Neumann boundary condition (14c) is applied over the rest of the boundary curve, where $θ∈[β2,β1]$⁠. Note here the dependence again on the body rotation due to the definition of $∂θ/∂x$ by (18c) in the current setting. Once again, practically, the above adaptation amounts to updating the vector of constants $b$ in our linear system. As a note of computational caution, when the body rotates, $β1$ and $β2$ may cross the point of $θ=0$ requiring careful implementation to ensure the conditions are applied correctly.

The computational domain and finite difference equations can now be formulated as in Smith and Liu,¹ taking care to introduce the new discretization coefficients for the directional body derivatives. The domain is discretized into an $M×N$ elliptical grid with $Δr=1/M$ and $Δθ=2π/N$⁠, where $ri=iΔr$⁠, $θj=jΔθ$ and $i∈[1,M]$⁠, $j∈[1,N]$⁠, thus excluding the origin. At each time step, the equations of motion are solved as a linear system using an $(M×N)×(M×N)$ finite difference matrix to model the body trajectory over time. To emphasize, the novel extension herein lies in the modeling of a freely pitching and rotating elliptical body without the need for a moving mesh or grid interpolation.

In the proceeding analysis, we (i) consider the effect of body rotation and ellipticity on the fluid–body interaction, (ii) study the influence that moving the body's center of mass has on stabilizing or destabilizing the interactions, and (iii) analytically consider the behavior of the body and fluid in a high-frequency regime.

The values of the underbody shape constants are taken to be $a,b,c=0.8,0.1,0.1$ producing a hump of height $c2=0.01$⁠. Note the scaling of a and b in (20) with q to ensure the hump's base remains within the planform boundary.

Furthermore, a grid independence study was carried out ahead of obtaining the presented results to ensure an accurate mesh resolution was used. The following results are performed on a $400×400$ grid, uniformly discretizing the polar domain of $r∈(0,e]$ and $θ∈(0,2π]$⁠.

Beginning now with the case of a circular planform, $q=1$⁠, we investigate the influence of rotation about the z-axis on the body's motion, in particular with regard to its height, $h(t)$⁠, in the flow and pitch, $θ1(t)$⁠. Figure 2 shows the evolution of these two quantities over time for different values of angular velocity $ω=0, π/2, π, 2π, 4π, 10π$⁠. In both top (h) and bottom (⁠ $θ$⁠) panels, the respective values can be seen to tend to a limit with increasing $ω$⁠. The insets in each show similar values for $2π,4π,10π$⁠. The results show increased stability in the body motion with delayed blow up/fly away in the motion as shown by a reduced height and pitch at time $t=10$⁠. The existence of a limit here indicates that the gyroscopic stability provided by rotation in the z-axis is of a finite influence with increased $ω$⁠. Notably, there is a small correction for higher frequencies with the solution for $ω=4π$ slightly “overshooting” that of $ω=10π$⁠. A dotted black line in Fig. 2 shows the formal limit, the existence and nature of which are discussed in Sec. IV B, which also shows a slight correction back from the highest frequency solutions.

Investigating further, Fig. 3 displays the effect of body rotation on both the lift force acting on the body (green curve, left axis) and the moment (purple curve, right axis) at time $t=10$⁠. With increased $ω$⁠, the forces acting over the body reach a limit around $ω=2π$⁠, demonstrating why little further variation in the body motion is seen for larger angular velocities since the body forces remain relatively similar.

A final point of interest in the circular planform case is the fluid pressure (p), and horizontal and vertical velocity components (u and v respectively) over the underbody. Figure 4 presents the results for these three quantities in turn at different radii on the body $r=0.33,0.66,0.99$⁠. The top plot shows the pressure, while the bottom left and right show u and v, respectively. Interestingly, there is a scaling factor of $1/7$ between the $ω=0$ (black curves, left axes) case and the $ω=10π$ (purple curves, right axis) case causing the profiles for each case and radii to match exactly.

Hence, the average body only varies with r, taking the same cross-sectional form for all $θ$⁠. Solving the equations of motion (14) with this new body function and $ω=0$⁠, we obtain the limiting curves for h and $θ$ in Fig. 2 that closely match the high angular velocity computational cases. The three-dimensional profile of this shape and consequent fluid pressure over its surface are presented in Fig. 5 wherein the results again match very closely across the whole body shape between the averaged body and rotating body cases.

For a final comparison, the fluid pressure (p) and horizontal and vertical velocity components (u and v respectively) over the underbody at different radii on the body $r=0.33,0.66,0.99$ are presented in Fig. 6, again showing good agreement between the limiting averaged body case and the rotating body.

In (15a) and (15c), the location of the body's center of mass plays a role in determining the evolution of its pitch within the flow and its roll (however, the latter is negligible in the overall body motion). Again for a circular planform, we compare the influence of $xc$ on a non-rotating body to one rotating with $ω=10π$ (left column and right column). Figure 7 shows the evolution of the body height h and pitch $θ1$ over time. For the non-rotating body, $xc>0$ can have a strong destabilizing effect blowing up the solutions for $h(t)$ and $θ1(t)$ with the magnitudes of both h and $θ1$ increasing faster as $xc$ approaches 1. Alternatively, a stabilizing effect is seen for $xc<0$⁠, with h and $θ1$ remaining small.

The opposite trend is seen for a body that rotates, although the magnitudes of h and $θ1$ are several orders smaller. Indeed, for $xc=1$ a curious interaction occurs in which the body trajectory reverses, decreasing h and increasing $θ1$ over the $t=[0,10]$ interval. Overall, rotation serves to provide stability for each value of $xc$⁠.

Similar trends are seen in the pressure under the body in Fig. 8 where p at a radius $r=0.99$ is shown for both the non-rotating and high-frequency rotation cases (top and bottom, respectively) at $t=10$⁠. Indeed, the magnitude and size ordering of the pressure curves follow that of Fig. 9, with greatly diminished pressures for $xc<0$ and when rotating. The link between the trends in p, h, and $θ1$ comes from the body motion Eq. (15), Eq. (15c) wherein it is the pressure over the underbody that governs the overall motion and coupled interaction.

One point of further interest is how the hump location (a, b, c) and $xc$ interact. The above results relate to a body with $a=0.8,b=0.1,c=0.1$⁠: see (20). Producing the same analysis for $a=−0.8,b=−0.1$⁠, as in Fig. 9 it can be seen that the aforementioned trends reverse. That is, when the hump and center of mass are located on the same side of the body (i.e., a and $xc$ are of the same sign), they serve to destabilize the body in comparison to when they are on opposite sides.

The final case we consider is that of bodies with varying elliptical planforms and the effect of rotation on their motion. In Fig. 10, the variations in body height and pitch over time are shown for different elliptical bodies. Once again, the left column depicts non-rotating bodies and the right column shows bodies rotating with velocity $ω=10π$⁠. As a reminder, the parameter q is the ratio of the body's x-axis to the body's y-axis, with circular bodies, $q=1$⁠, depicted for comparison (yellow curves). In each variation of q, the position of the hump location is scaled accordingly with $a=0.8/q$ and $b=0.8q$ to ensure that it remains within the planform.

For the non-rotating case, $q>1$⁠, the body's y-axis is larger than the x-axis, hence the body is wide in the spanwise direction and thin in the streamwise direction. Treating the circular body $q=1$ as a reference case, for $q>1$ the body is seen to interact more strongly with the flow, moving away from the wall (h) and pitching at a faster rate. The opposite is seen for $q<1$ where the body is spanwise thin and streamwise thick. The body trajectory, according to h, is now much shallower throughout and evolves more slowly.

Under rotation, two properties of interest are seen. First, $q=1$⁠, the circular case is now the fastest growing solution, however the cases for other q values remain of similar order. Second, the cases for $q=0.4$ and 1.6 produce similar body trajectories, as do $q=0.7$ and 1.3 since the thicknesses of the body in spanwise and streamwise directions continually vary becoming periodically thick and thin. Thus, these pairings of q values present similarly sized and averaged body shaped bodies when rotating. The similarity in these results is further emphasized by Fig. 11 in which the body height (left axis, green) and pitch (right axis, purple) are shown for different q at $t=10$⁠. For the non-rotating case, the circular body sits between the results for $q<1$ and $q>1$⁠, with the magnitude of the lift force and moment both increasing with q. Under rotation, the circular case is seen to be the peak value with rough symmetry either side for high and low q. We thus anticipate that if $q=q̂$⁠, where $q̂$ is some real scalar, under rotation the body trajectory results would match the case where $q=1/q̂$⁠.

In this paper, we have studied the three-dimensional fluid–body interaction of a freely moving, submerged body in a near-wall flow. Building upon the work of Smith and Liu,¹ we extended their computational framework to include a rotating body in order to study the effect of a body's rotation about its vertical axis, perpendicular to the wall, on its motion over time. To this end, we used a fixed-frame finite difference method to efficiently solve the equations of motion and produce novel studies into how a body's rotation, its ellipticity and its center of mass location affect its motion. We demonstrated previously unseen (within this context) stabilization or destabilization of the body over time. The inclusion of a non-smooth underbody added further novelty to the results and study. In this setting, and through a series of different computational analyses, three key findings have emerged.

First, the high-frequency rotation of a circular body stabilizes its motion through the fluid (i.e., delays and slows its fly away and pitching) toward a finite limit, that of a non-rotating rotationally averaged underbody. As such, the high-frequency solution is independent of the angular velocity above a certain threshold. From Sec. IV B and Figs. 2, 6, and 5 it is apparent that this gyroscopic effect comes into play for large $ω$ essentially scaling the body variables found for $ω=0$ by a factor 1/7 (for our prescribed g). For a rotationally averaged body shape $gav$ (that does not rotate, i.e., $ω=0$⁠), this gyroscopic stabilization is built into the underbody shape. The reasoning behind this phenomenon can be observed in (14a) due to the dependence on $∂f/∂x$ on the right-hand side. Since the contribution of $∂f/∂x$ depends on the body configuration and alignment in the flow, when the timescale of body rotation is fast compared to the evolution of its height and pitch (i.e., large $ω$⁠), the fluid essentially only “sees” the rotationally averaged body and its averaged derivative. Finally, we note that the plateau of the stabilization effect at high rotational frequencies may well be due simply to the signs of the small perturbations about the limit.

Second, the body motion is greatly affected by the center of mass location, leading to a blow up in the body height and pitch of several orders for non-rotating bodies. Under rotation, the center of mass and underbody hump show further interaction such that when located on the same side the body motion is greatly diminished, even showing a slight reversal in direction (decreasing h and increasing $θ$⁠), compared to the case when both hump and center of mass are on opposite sides causing a mild destabilization (in comparison to a centrally located center of mass). Indeed, throughout all of the analyses in this paper the body is shown to fly away, i.e., move away from the wall, in almost all cases, except for one scenario. When the body is rotating and the center of mass is ahead of the hump, $xc>a>0$⁠, the body moves toward to wall. This is highlighted in Fig. 9 wherein the mirrored scenario (⁠ $xc<a<0$⁠) is shown to match and show the same destabilization. In these instances, the center of mass and hump interact to “over-stabilize” the body, reversing its typical trajectory. When the body moves closer to wall, we anticipate that viscous effects due to the non-slip wall condition will emerge, requiring further analysis as to whether impact ensues, as considered in several two-dimensional analyses.^{12,25,29,33–35}

Third, varying the ellipticity of the body leads to further changes in the body trajectory. For bodies of greater spanwise than streamwise thickness, the magnitude of both h and $θ1$ increases faster than in the opposite case. Hence, for a non-rotating body, the streamwise and spanwise presentation (shape) of the body to the fluid affect the time until the body flies way. When q is small, its streamwise dimension is greater than its spanwise dimension and serves to stabilize the body in the flow; while, for large q the opposite occurs. When the body rotates, the differences in the body's relative spanwise and streamwise extent average out and all body shapes remain similarly stable in the flow. This is further seen since, under rotation, results for $q=q̂$⁠, where $q̂$ is some real scalar, showing agreement with the case where $q=1/q̂$ (up to the location of the hump), e.g., on average the fluid “sees” the same body shape for small and large q.

Thus, overall, it is seen that (1) body rotation has a generally favorable effect causing a body to stabilize and thus maintain proximity to a wall for longer through gyroscopic effects, (2) for elliptical bodies, the eccentricity of the body has very limited effect on the body trajectory when rotating, especially in comparison to the non-rotating case in which more significant fly away and blow up motions are seen, and (3) the location of the center of mass quantitatively and qualitatively affects the body trajectory, with body rotation reversing the trends seen in the non-rotating case for different values of $xc$⁠. The novelty of this study in including these rotational body dynamics and its present three-dimensional setting enable the evaluation of the above phenomena.

There are several limitations to note with this modeling framework. First, the assumption of inviscid flow combined with a thin attached viscous three-dimensional boundary layer on the solid surface might limit the practical applications of this work. Since this study is an initial attempt at modeling such three-dimensional interactions, this assumption is reasonable to enable theoretical progress.

Second, we have assumed the flow remains attached throughout the interaction. The relevance of this assumption depends on the nature of the boundary layer. If the viscous boundary layer is predominantly turbulent, then the attached flow assumption is largely valid at high Reynolds numbers. If the boundary layer is predominantly laminar, then attached flow usually still applies for a finite scaled time before detachment and large-scale separation take place. These two points require further investigation and form potentially fruitful directions for future work. Indeed, a separated inviscid flow model could be built upon the present attached version and in principle to describe the separated flow scenario once separation is fairly widespread.

Third, further regarding the applicability of this method to real-world scenarios, a comparison to experimental data or direct numerical simulations of this interaction would be valuable for validating the results; however, such data and simulations are currently unavailable to us.

Fourth, our computational method is fast and efficient for the current setting; however, it is likely to lack flexibility and applicability for other settings. Indeed, we have presented results for the linearized problem and as such our proposed method is likely to be unsuitable for more complex scenarios, e.g., the inclusion of viscous effects and where a Poisson equation no longer holds in the wall gap. For such scenarios, we anticipate that different numerical methods and frameworks will be required.

In light of these limitations we anticipate several fruitful directions for future work, which include (1) three-dimensional formulation and body rotation effects in different flow settings of fluid–body interactions, such as in boundary layers and channel flows, and different scalings of the body relative to the wall, (2) further numerical study of this setting using finite element methods and direct numerical simulation as well as comparable experimental results to validate the applicability of the developed theory, (3) a non-linear version of the interactive system would be of value wherein it is expected that the roll motion of the body becomes important, (4) extension of fluid–body skimming interactions, particularly,^25,28,36 and (5) an exploration of this model's relevance to real-world applications. This includes its applicability to aerospace and automotive engineering, particularly the movement of ice particles, droplets, and debris within a vehicle's boundary layer; underwater robotics and the motion of bodies close to the seabed or boundary layer under ice floes; and ecological and environmental phenomena such as pollen and agrochemical transport in environmental flows. This last case may also feature multiphysical effects such as electrostatics^37–39 that will serve as a novel extension of fluid–body interaction problems.

The authors have no conflicts to disclose.

Ryan A. Palmer: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Kevin Liu: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Frank T. Smith: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.