Direct numerical simulation of the oscillatory flow around a sphere resting on a rough bottom

TL;DR

This study uses direct numerical simulations to analyze oscillatory flow around a sphere on a rough bottom, revealing vortex shedding, turbulence generation, and sediment interaction relevant to marine environments.

Contribution

It introduces a detailed simulation approach for flow around a sphere on a rough seabed, including sediment particles, to study vortex dynamics and sediment transport.

Findings

Vortex structures can break up and generate turbulence at high Reynolds numbers.

Flow velocity fields enable analysis of vortex dynamics and turbulence.

Forces and torques on sediment particles and the sphere are quantifiable.

Abstract

The oscillatory flow around a spherical object lying on a rough bottom is investigated by means of direct numerical simulations of continuity and Navier-Stokes equations. The rough bottom is simulated by a layer/multiple layers of spherical particles, the size of which is much smaller that the size of the object. The period and amplitude of the velocity oscillations of the free stream are chosen to mimic the flow at the bottom of sea waves and the size of the small spherical particles falls in the range of coarse sand/very fine gravel. Even though the computational costs allow only the simulation of moderate values of the Reynolds number characterizing the bottom boundary layer, the results show that the coherent vortex structures, shed by the spherical object, can break-up and generate turbulence, if the Reynolds number of the object is sufficiently large. The knowledge of the velocity field allows the dynamics of the large scale coherent vortices shed by the object to be determined and turbulence characteristics to be evaluated. Moreover, the forces and torques acting on both the large spherical object and the small particles, simulating sediment grains, can be determined and analysed, thus laying the groundwork for the investigation of sediment dynamics and scour developments.

Tables

Table 1. Table 1: Summary of domain discretization and flow parameters for the present runs.

$R_{\delta }$	$D$	$d$	no. layers of spheres	$ϵ$	$L_{x_1}$	$L_{x_2}$	$L_{x_3}$	$\Delta x_{fine}$	$\Delta x_{coarse}$	$\frac{D}{\Delta x_{fine}}$	no. small spheres
$47.87$	$6.267$	$-$	$-$	$0.09777$	$60.54$	$60.54$	$30.27$	$0.059$	$0.24$	$106$	$-$
$56.0$	$28.0$	$2.6$	$1$	$0.374$	$191.3$	$95.66$	$95.66$	$0.19$	$0.75$	$150$	$2338$
$112.1$	$28.0$	$2.6$	$1$	$0.374$	$382.6$	$95.66$	$95.66$	$0.19$	$0.75$	$150$	$4740$
$112.1$	$28.0$	$2.6$	$5$	$0.374$	$382.6$	$95.66$	$95.66$	$0.19$	$1.50$	$150$	$24548$
$112.1$	$11.2$	$0.56$	$3$	$0.112$	$172.2$	$43.05$	$43.05$	$0.056$	$0.45$	$200$	$56620$

Equations

$$\frac{\partial p^{\ast }}{\partial x_1^{\ast }}=-{\rho }^{\ast }{U}_0^{\ast }{\omega }^{\ast }\sin ({\omega }^{\ast }{t}^{\ast });\frac{\partial p^{\ast }}{\partial x_2^{\ast }}=0;\frac{\partial p^{\ast }}{\partial x_3^{\ast }}=0$$

$$t=t^{\ast }{\omega }^{\ast };(x_1,x_2,x_3)=\frac{(x_1^{\ast },x_2^{\ast },x_3^{\ast })}{{\delta }^{\ast }};(u_1,u_2,u_3)=\frac{(u_1^{\ast },u_2^{\ast },u_3^{\ast })}{U_0^{\ast }};p=\frac{p^{\ast }}{{\rho }^{\ast }(U_0^{\ast }{)}^2}$$

$$\frac{\partial u_j}{\partial x_j}=0$$

$$\frac{\partial u_i}{\partial t}+\frac{R_{\delta }}{2}\left(u_j\frac{\partial u_i}{\partial x_j}\right)=-\frac{R_{\delta }}{2}\frac{\partial p}{\partial x_i}-{\delta }_{k1}\sin (t)+\frac{1}{2}\left(\frac{{\partial }^2{u}_i}{\partial x_k\partial x_k}\right)+f_i$$

$$R_{\delta }=\frac{U_0^{\ast }{\delta }^{\ast }}{{\nu }^{\ast }}$$

$$\left(\frac{\partial u_1}{\partial x_3},\frac{\partial u_2}{\partial x_3}\right)=(0,0);u_3=0\text{mbox}at{x}_3=L_{x_3}$$

$$(u_1,u_2,u_3)=(0,0,0)\text{mbox}at{x}_3=0$$

$$F_i=\frac{8F_i^{\ast }}{{\rho }^{\ast }{\left(U_0^{\ast }\right)}^2\pi d^{\ast 2}}=\frac{8}{\pi d^2}{\int }_S{t}_{i}dS,T_i=\frac{16T_i^{\ast }}{{\rho }^{\ast }{\left(U_0^{\ast }\right)}^2\pi d^{\ast 3}}=\frac{16}{\pi d^3}{\int }_S{\delta }_{ijk}{x}_j{t}_{k}dS,$$

Full text

1   Introduction

An object lying on the sea bed causes a local acceleration of the flow field and a local increase of the bottom shear stress. It follows that the sediments surrounding the object might be swept away from it, causing a local lowering of the bed profile, even when the flow far from the object is not strong enough to move the sediment. This phenomenon is observed at different spatial scales, which range from that of a small pebble lying on a sandy bottom to that of the foundation of a large coastal structure. Then, the scour which develops around the object has different consequences as, for example, the self-buring of the object (e.g. self-burial of a pipeline) and its possible instability (e.g. the instability of a wind mill). The self-buring of the object is also quite important for mines or unexploded ordnance (UXO). For example, nowdays, because of increasing human activities in shallow waters (e.g. navigation, fisheries, sand extraction), the buried mines and UXO of World Wars I and II are a threat to public safety and remediation in many coastal areas is becoming a priority.

A key role in the mechanics of sediment transport and in the dynamics of the scour around the object is played by the dynamics of the vortices which are generated by the interaction of the external flow with the object. Indeed, these vortices tend to pick-up the sediment grains from the bottom, to make them rolling and sliding along the bottom surface or even to carry them into suspension, when the vertical velocity they induce is larger than the fall velocity of the sediment particles.

The determination of the threshold conditions, above which the sediments start to move or are carried into suspension, is a complex problem. Indeed, to quantify the bottom erodibility, it is necessary to know both the mechanical properties of the sediments (e.g. grain size and sediment density) and the dynamics of the large vortex structures which are present close to the bottom and can induce, locally, large values of the hydrodynamic forces acting on sediment particles.

In coastal environments, the phenomenon is made more complex by the oscillatory character of the flow induced by the propagation of sea waves, which makes the vortex structures shed by the object during a half cycle to interact with the object and the vortices shed during the previous half cycle. This nonlinear interaction might give rise to a possible chaotic flow which might appear through different scenarios, e.g. Feigenbaum scenario (Blondeaux & Vittori, 1991) and quasi-periodicity and phase-locking scenario (Vittori & Blondeaux, 1993).

Moreover, for relatively small objects, the vortices shed by the object might interact with the small eddies shed by the sediment grains and the eddies generated by the transition process from the laminar to the turbulent regime in the bottom boundary layer. On the other hand, for relatively large objects, the vortices shed by the object do not interact directly with the small eddies shed by the sediment grains and the turbulent eddies, even though the latter certainly affect the dynamics of the former.

Therefore, to obtain an accurate and reliable description of the phenomenon, it is necessary to consider the simultaneous presence of i) the vortex structures shed by the object, ii) the small vortices shed by the sediment grains and iii) the possible presence of turbulent eddies. The turbulent eddies appear when the Reynolds number of the flow is large enough to trigger the transition process from the laminar to the turbulent regime either in the bottom boundary layer or in the free shear layers released by the object invested by the oscillatory flow.

For an oscillatory boundary layer over a smooth wall, both experimental measurements (e.g. Hino et al., 1976) and direct numerical simulations (e.g. Verzicco & Vittori, 1996; Vittori & Verzicco, 1998) indicate that turbulence appears explosively during the decelerating phases of the oscillatory cycle, when the Reynolds number $R_{\delta}$ is larger than a value ranging between $500$ and $600$. Hereinafter, the Reynolds number is defined with the amplitude $U^{*}_{0}$ of the velocity oscillations far from the bottom and the thickness of the viscous bottom boundary layer $\delta^{*}=\sqrt{2\nu^{*}/\omega^{*}}$, $\nu^{*}$ being the kinematic viscosity of the water and $\omega^{*}$ the angular frequency of the velocity oscillations. However, close to the critical conditions, turbulence does not survive during the accelerating phases and the flow recovers a laminar ’like’ behaviour (’intermittently turbulent regime’). Larger values of the Reynolds number cause turbulence to appear earlier and to pervade larger parts of the cycle till, at high Reynolds numbers, turbulence is present throughout the cycle. The direct numerical simulations of Costamagna et al. (2003) showed that the elementary process which generates turbulent eddies in an oscillatory flow is similar to that in a steady flow.

More recently, Carstensen et al. (2010) observed the presence of turbulent spots during the transition process from the laminar to the turbulent regime in an oscillatory boundary layer and showed that these isolated turbulent areas in an otherwise laminar boundary layer, cause violent oscillations of both the velocity and the shear stress. Moreover, Carstensen et al. (2010) observed that turbulent spots emerge from the breaking of low speed streaks. Later, the existence of turbulent spots in an oscillatory boundary layer was confirmed by the direct numerical simulations of Mazzuoli et al. (2011). Since the numerical simulations give access to velocity and pressure fields in the three-dimensional space and time, the numerical results of Mazzuoli et al. (2011) supplemented the experimental measurements of Carstensen et al. (2010) and in particular allowed to determine the speed of the head and tail of the spots along with the speed of the lateral spreading of the spot.

As already pointed out, the studies summarized so far were carried out by considering a smooth bottom. In natural environments, the sediment grains make the bottom to be rough and generate small vortices, which interact with the turbulent eddies. An experimental investigation of the oscillatory flow over macro-roughness elements was made by Sleath (1976), who measured the velocity profile in an oscillatory boundary layer over spheres of large diameter, arranged in an hexagonal pattern. The measurements of Sleath (1976) showed a complex turbulent flow field and suggested the existence of coherent vortex structures which are shed by the roughness elements at flow reversal and move away from the bottom. The oscillatory flow over a similar rough bottom was investigated by Fornarelli & Vittori (2009) by means of direct numerical simulations of continuity and Navier-Stokes equations. The roughness consisted of semi-spheres regularly fixed on a plane wall in an hexagonal pattern. The results of Fornarelli & Vittori (2009) show that the temporal development of the velocity close to the spheres is characterized by two maxima. One maximum is correlated to the maximum of the free stream velocity. A further peak in the velocity appears close to the reversal of the external flow and is generated by the passage of the vortex structures shed by the roughness elements, which move away from the bottom.

More recently, the oscillatory flow over a layer of spherical grains has been simulated by Mazzuoli & Vittori (2016) for different values of the diameter of the spheres and different values of the Reynolds number. Their results show that at least three flow regimes exist, namely the laminar regime, the transitional turbulent regime and the hydrodynamically rough turbulent regime. For relatively small values of the Reynolds number, the turbulent kinetic energy has negligible values and the flow regime can be defined laminar. When the Reynolds number is increased, two different regimes are encountered depending on the value of the sphere size. For small spheres, it is likely that the turbulent regime is due to an intrinsically instability of the oscillatory boundary layer. Indeed, turbulent fluctuations are observed when the Reynolds number is larger than a critical value similar to that of the Stokes boundary layer over a flat wall. On the other hand, for large spheres, turbulence is generated by the nonlinear interaction of the free shear layers shed by the sediment grains.

There are many other results on the flow over a bottom of regular roughness elements. For example, let us mention that, recently, Celik et al. (2014) have carried out pressure measurements on a spherical grain resting upon a bed of identical grains. However, even though interesting results have been obtained by Celik et al. (2014), their results as well as other results which are not summarized herein consider a steady forcing flow, the characteristics of which are different from those of an oscillatory flow.

Much less is known on the dynamics of the three-dimensional vortex structures shed by an object lying on a flat wall and subject to an oscillatory flow. Fischer et al. (2002) made direct numerical simulations of the oscillatory flow around a sphere laying on a plane wall. The investigation of Fischer et al. (2002) was inspired by the experiments described by Rosenthal & Sleath (1986) and was aimed at providing more information on the lift and drag forces acting on a sediment grain at the bottom of sea waves. Unlike the results of Cherukat & McLaughlin (1994), Cherukat et al. (1994), Asmolov (1999) and Asmolov & McLaughlin (1999), the numerical result of Fischer et al. (2002) are not restricted to relatively small values of the Reynolds number or to disparate diffusive, convective and oscillatory length scales. However, attention was focused on the forces acting an the sphere and the velocity and vorticity field around the sphere as well as the shear stress acting on the bottom were not analysed.

Let us mention also the recent studies of the flow and scour around pipelines and piles of Fuhrman et al. (2014) and Baykal et al. (2015) where the results of previous studies are also summarized. The investigations of Fuhrman et al. (2014) and Baykal et al. (2015) were carried out by solving the Reynolds averaged Navier-Stokes (RANS) equations and introducing a two-equation turbulence model. The introduction of the Reynolds average has the advantage of allowing the simulation of flow fields characterized by high Reynolds numbers, however, a RANS approach does not resolve explicitly the turbulent mixing and hence does not provide accurate results about the dynamics of the coherent vortices shed by the objects and their interaction with the turbulent eddies and the vortices shed by the sediment grains.

The present paper describes the results of direct numerical simulations of the oscillatory flow over an idealized sea bottom made by small spherical particles, which simulate a coarse sand or a very fine gravel sediment, above which a much larger sphere is resting, which can be though to represent any object at rest (e.g. a small cobble or a small unexploded ordnance). The use of direct numerical simulations allows us to evaluate quantities that are very difficult to measure in a laboratory experiment (e.g. vorticity, dissipation and production of turbulence, …). Moreover, the results of direct numerical simulations allow a detailed study of the vortex structures generated during the oscillatory cycle to be carried out, along with the investigation of the interaction of the vortices with the particles lying on the bottom.

The forcing flow is assumed to be oscillatory and generated by the propagation of a surface wave. Hence, the frequency of the fluid oscillations is chosen to reproduce what happens at the bottom of a sea wave. The computational costs do not allow high Renyolds numbers to be simulated but results for values of the Reynolds number large enough to trigger transition to turbulence are obtained and presented.

The structure of the rest of the paper is the following. In the next section, we formulate the problem and summarize the main steps of the numerical procedure employed to determine the oscillatory flow around a spherical object resting on the sea bed. In section 3, we describe the results focusing the attention on the dynamics of the coherent vortex structures shed by the object and on their interaction with the bottom and the roughness elements. Section 4 for is devoted to the conclusions and to a brief description of the future developments of the work.

2   Formulation of the problem and numerical approach