Research Interests
Computational fluid dynamics; fluid/solid interactions; applications of fluid dynamics in biological applications; high performance computing (MPI, CUDA C).
Computational fluid dynamics; fluid/solid interactions; applications of fluid dynamics in biological applications; high performance computing (MPI, CUDA C).
Numerical study of the material transport in the viscous vortex dipole flow
(2019) Physics of Fluids, 31: 053602.
Ling Xu SUMMARY: This paper presents a numerical study of the material transport of Lamb dipole(s) in the twodimensional viscous flow. We focus on the properties of the rate of strain tensor, which has received less attention in the literature. It is noted that the eigenpairs of the tensor explicitly indicate the strength and direction of material stretching and compressing. The tensor provides a clear map of the material motion regardless of the complexity of the vortical flow. The strain rate field displays a rich structure as it contains five elliptic points and six hyperbolic points. It is interesting to observe that the left elliptic point of the strain rate field bifurcates into two at t > 0. Two kinds of material curves, circular and vertical, are used to illustrate the flow transport. The transport mechanism discussed here can be employed to explore the transport in more complex vortex flows. KEY WORDS: transport,strain rate, Lamb dipole, bifurcation, critical points, viscous flow 
Figure: Lamb dipole, (a) color plot of strain rate γ, (b) critical points of γ, elliptic points (E_L, E_R, E_T , E_C, E_B), hyperbolic points (H_NW , H_NE, H_SW , H_SE, H_T , H_B), subscript indicates location, also shown are several contours of γ. 

Computation of the starting vortex flow past a flat plate
(2017) Procedia IUTAM, 136143.
Ling Xu, Monika Nitsche, Robert Krasny SUMMARY: This paper compares two numerical methods applied to compute the starting vortex flow past a flat plate. The plate is inclined relative to a constant background flow at angle α, with α = 90◦ , 60◦ , 30◦. The numerical methods considered are (1) direct numerical simulation of the viscous flow (DNS), and (2) an inviscid vortex sheet model. The viscous DNS solves the Navier Stokes equations by an operator splitting finitedifference method, for Reynolds numbers Re = 250, 500, 1000, 2000. The inviscid flow is computed by a regularized vortex sheet method, with the unsteady Kutta condition imposed at the edges of the plate, for regularization parameters δ = 0.2, 0.1, 0.05. We present viscous vorticity contours, and compare streaklines and shed circulation obtained with both methods. Good agreement is found in the largescale features of the separated spiral streaklines and the shed circulation as Re increases and δ decreases. For small inclination angle α, secondary separation on the downwind side of the plate introduces smallscale features in the viscous flow that are absent in the inviscid model. The vortex sheet model is much less costly than the viscous DNS, but it is limited by the omission of the boundary layers present in the viscous flow. KEY WORDS: Starting vortex flow, separation, NavierStokes, vortex sheet model, unsteady Kutta condition 
Figure: Streaklines obtained from DNS with Re = 500 at the indicated times, for α = 90◦ (left), 60◦ (middle), 30◦ (right). 

Numerical study of viscous starting flow past a wedge
(2016) Journal of Fluid Mechanics, 801:150165.
Ling Xu SUMMARY: This paper presents a numerical study of vortex formation in the impulsively started viscous flow past an infinite wedge, for wedge angles ranging from 60∘ to 150∘. The Navier–Stokes equations are solved in the vorticitystreamfunction formulation using a timesplitting scheme. The vorticity convection is computed using a semiLagrangian method. The vorticity diffusion is computed using an implicit finite difference scheme, after mapping the physical domain conformally onto a rectangle. The results show details of the vorticity evolution and associated streamline and streakline patterns. In particular, a hierarchical formation of recirculating regions corresponding to alternating signs of vorticity is revealed. The appearance times of these vorticity regions of alternate signs, as well as their dependence on the wedge angles, are investigated. The scaling behaviour of the vortex centre trajectory and vorticity is reported, and solutions are compared with those available from laboratory experiments and the inviscid similarity theory. KEY WORDS: boundary layer separation; coastal engineering; vortex dynamics Figure: Evolution of the vorticity (left), streamlines (middle), and streaklines (right) at a sequence of times. 


Cilium height difference between strokes
is more effective in driving fluid
transport in mucociliary clearance: a numerical study
(2015) Mathematical Biosciences and Engineering, 12(5).
Ling Xu and Yi Jiang. SUMMARY: Mucociliary clearance is the first line of defense in our airway. The purpose of this study is to identify and study key factors in the cilia motion that influence the transport ability of the mucociliary system. Using a rod propelfluid model, we examine the effects of cilia density, beating frequency, metachronal wavelength, and the extending height of the beating cilia. We first verify that asymmetry in the cilia motion is key to developing transport in the mucus flow. Next, two types of asymmetries between the effective and recovery strokes of the cilia motion are considered, the cilium beating velocity difference and the cilium height difference. We show that the cilium height difference is more efficient in driving the transport, and the more bend the cilium during the recovery stroke is, the more effective the transport would be. It is found that the transport capacity of the mucociliary system increases with cilia density and cilia beating frequency, but saturates above by a threshold value in both density and frequency. The metachronal wave that results from the phase lag among cilia does not contribute much to the mucus transport, which is consistent with the experimental observation of Sleigh (1989). We also test the effect of mucus viscosity, whose value is found to be inversely proportional to the transport ability. While multiple parts have to interplay and coordinate to allow for most effective mucociliary clearance, our findings from a simple model move us closer to understanding the effects of the cilia motion on the efficiency of this clearance system. KEY WORDS: Mucus, cilium, transport, propulsion, NavierStokes 
Figure: (a) Streamlines induced by five rods at a sequence of times, as indicated in the plots. The rod length shortens. (b) Streaklines of fluid particles at a sequence of times. (c1) Seven cases of the rod length Lc are plotted as a function of its orientation θ. The solid line is the length for the effective stroke, and the dash lines are for the recovery stroke. (c2) Compare the fluid particle displacements D of curves 1 to 5 in (c1). (c3) Compare the fluid particle displacements D of curves 3, 6 and 7 in (c1). The displacement D is plotted as a function of tp, the time when the particle is released into fluid. For all plots of this figure, the number of rods Nrod = 5, the beating frequency is the same between strokes, f = 15.625Hz, and the phase shift among rods is φ = 0.2. 

Startup vortex flow past an accelerated flat plate
(2015) Physics of Fluids, 27: 033602.
Ling Xu and Monika Nitsche. SUMMARY: Viscous flow past a finite flat plate accelerating in the direction normal to itself is studied numerically. The plate moves with nondimensional speed tp, where p = 0,1/2,1,2. The work focuses on resolving the flow at early to moderately large times, and determining the dependence on the acceleration parameter p. Three stages in the vortex evolution are identified and quantified. The first stage, referred to as the Rayleigh stage (Luchini & Tognaccini 2002), consists of a vortical boundary layer of roughly uniform thickness surrounding the plate and its tip, without any separating streamlines. This stage is present only for p > 0, for a time interval that scales like p^3, as p goes to 0. The second stage is one of selfsimilar growth. The vortex trajectory and circulation satisfy inviscid scaling laws, the boundary layer thickness satisfies viscous laws. The selfsimilar trajectory starts immediately after the Rayleigh stage ends, and lasts until the plate has moved a distance d = 0.5 to 1 times its length. Finally, in the third stage, the image vorticity due to the finite plate length becomes relevant and the flow departs from selfsimilar growth. The onset of an instability in the outer spiral vortex turns is also observed, however, at least for the zerothickness plate considered here, it is shown to be easily triggered numerically by underresolution. The present numerical results are compared with experimental results of Pullin & Perry (1980), and numerical results of Koumoutsakos & Shiels (1996). KEY WORDS: Starting vortex; power law; viscous flow; s eparation; Reynolds; streaklines; vortex center Figure: Left column: Streaklines for flow past a wedge of angle β = 5 degrees, at t=1s, 1.6s, 2.8s, 4s and 5s, for p = 0.45 and Re = 6621, obtained by Pullin and Perry (1980) from laboratory experiments (reproduced with permission from the Journal of Fluid Mechanics). Middle and Right columns: Numerical simulations of streaklines and streamlines, respectively, for flow past a flat plate at the same times, for p = 0.45 and Re = 6000. 

Scaling behaviour in impulsively started viscous
flow past a finite flat plate
(2014) Journal of Fluid Mechanics, 756:689715.
Ling Xu and Monika Nitsche SUMMARY: Viscous flow past a finite flat plate which is impulsively started in the direction normal to itself is studied numerically using a high order mixed finitedifference and semi Lagrangian scheme. The goal is to resolve details of the vorticity generation, and to determine the dependence of the flow on time and Reynolds number. Vorticity contours, streaklines and streamlines are presented for a range of times t ∈ [0.0002, 5] and Reynolds numbers Re ∈ [250, 2000], nondimensionalized with respect to the driving velocity and the plate length. At early times, the starting vortex is small relative to the plate length and is expected to grow as if an external length scale were absent. We identify three different types of scaling behaviours consistent with this premise: At early times, (1) solutions with different values of Re are identical up to rescaling. (2) The solution for fixed Re satisfies a viscous similarity law in time, locally in space, as illustrated by the core vorticity maximum, the upstream boundary layer thickness, and the maximum speed, in three different regions of the flow. (3) The vortex core trajectory and the shed circulation satisfy inviscid scaling laws for several decades in time, and are consequently essentially Reindependent at these times. In addition, the computed induced drag and tangential forces are found to follow approximate scaling laws that define their dependence on time and Re. KEY WORDS: viscous scaling for variable Re; viscous scaling in time for fixed Re; inviscid scaling in time; Navier Stokes; vortex dynamics; Figure: Vorticity contours near the plate tip. (Left column): solution at a fixed time t = 0.005, for increasing values of Re. (Right column): solution for fixed Re = 500, at an increasing sequence of times. 


Blast Induced Rock Fracture Near a Tunnel
(2014)
International Journal for Numerical and Analytical
methods in Geomechanics, DOI: 10.1002.
Ling Xu, Howard Schreyer and Deborah Sulsky SUMMARY: Damage in the form of cracks is predicted to assess the susceptibility of a tunnel to failure due to a blast. The materialpoint method is used in conjunction with a decohesive failure model as the basis for the numerical simulations. The assumption of a cylindrical charge as the source for the blast allows the restriction of plane strain and twodimensional analyses. In the simulation, a further restriction of a single pressure pulse is used as the source of stress waves that are reflected and refracted after reaching the free surface of the tunnel wall. Three critical zones of significant cracking in the vicinity of a tunnel are identified as potential contributors to tunnel failure. KEY WORDS: blast wave simulation; materialpoint method; tunnel failure; decohesive model for rock failure 
Figure: Distribution and orientation of rock (granite) fractures near a tunnel wall due to a blast. The blast is modeled as a large compressive wave The tunnel is absent (left). A zoomin of the damage region when the tunnel is present (right). The color red indicates locations of large fractures. 

Circulation shedding in viscous starting flow past a flat plate
(2014) Fluid Dynamics Research, (6):061420.
Monika Nitsche and Ling Xu SUMMARY: Numerical simulations of viscous flow past a flat plate moving in direction normal to itself reveal details of the vortical structure of the flow. At early times, most of the vorticity is attached to the plate. This paper introduces a definition of the shed circulation at all times and shows that it indeed represents vorticity that separates and remains separated from the plate. During a large initial time period, the shed circulation satisfies the scaling laws predicted for selfsimilar inviscid separation. Various contributions to the circulation shedding rate are presented. The results show that during this initial time period, viscous diffusion of vorticity out of the vortex is significant, but appears to be independent of the value of the Reynolds number. At later times, the departure of the shed circulation from its large Reynolds number behaviour is significantly affected by diffusive loss of vorticity through the symmetry axis. A timescale is proposed that describes when the viscous loss through the axis becomes relevant. The simulations provide benchmark results to evaluate simpler separation models such as point vortex and vortex sheet models. A comparison with vortex sheet results is included. KEY WORDS: Circulation, vorticity separation, vortex sheet simulations. 
Figure: Sketch defining the domain for computing circulation shedding in a viscous flow at (a) early times, (b) intermediate times, (c) late times. Figure: Shed circulation of (a) viscous flow and (b) vortex sheet flow. 
