Shear-imposed falling film

2014 ◽  
Vol 753 ◽  
pp. 131-149 ◽  
Author(s):  
Arghya Samanta
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Travelling Waves ◽  
Wave Structure ◽  
Travelling Wave ◽  
Neutral Stability ◽  
Constant Shear Stress ◽  
Analytical Result ◽  
Low Dimensional ◽  
Primary Instability

AbstractThe study of a film falling down an inclined plane is revisited in the presence of imposed shear stress. Earlier studies regarding this topic (Smith, J. Fluid Mech., vol. 217, 1990, pp. 469–485; Wei, Phys. Fluids, vol. 17, 2005a, 012103), developed on the basis of a low Reynolds number, are extended up to moderate values of the Reynolds number. The mechanism of the primary instability is provided under the framework of a two-wave structure, which is normally a combination of kinematic and dynamic waves. In general, the primary instability appears when the kinematic wave speed exceeds the speed of dynamic waves. An equality criterion between their speeds yields the neutral stability condition. Similarly, it is revealed that the nonlinear travelling wave solutions also depend on the kinematic and dynamic wave speeds, and an equality criterion between the speeds leads to an analytical expression for the speed of a family of travelling waves as a function of the Froude number. This new analytical result is compared with numerical prediction, and an excellent agreement is achieved. Direct numerical simulations of the low-dimensional model have been performed in order to analyse the spatiotemporal behaviour of nonlinear waves by applying a constant shear stress in the upstream and downstream directions. It is noticed that the presence of imposed shear stress in the upstream (downstream) direction makes the evolution of spatially growing waves weaker (stronger).

scholarly journals Recurrence of travelling waves in transitional pipe flow

2007 ◽  
Vol 584 ◽  
pp. 69-102 ◽  
Author(s):  
R. R. KERSWELL ◽  
O. R. TUTTY
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Pipe Flow ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Transition Process ◽  
Travelling Wave ◽  
Shear Stresses ◽  
Wall Shear ◽  
Dividing Surface

The recent theoretical discovery of families of unstable travelling-wave solutions in pipe flow at Reynolds numbers lower than the transitional range, naturally raises the question of their relevance to the turbulent transition process. Here, a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface in phase space which separates initial conditions which uneventfully relaminarize and those which lead to a turbulent evolution. This dividing surface (a separatrix if turbulence is a sustained state) is then minimally the union of the stable manifolds of all these travelling waves. Evidence for recurrent travelling-wave visits is found in both 5D and 10D long periodic pipes, but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10% of the time in turbulent flow at Re = 2400. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion solely over the travelling waves in which their individual properties are appropriately weighted. Instead the onus is on isolating further dynamical structures such as periodic orbits and including them in any such expansion.

The dynamics of driven rotating flow in stadium-shaped domains

1995 ◽  
Vol 294 ◽  
pp. 47-69 ◽  
Author(s):  
J. J. Kobine ◽  
T. Mullin ◽  
T. J. Price
Keyword(s):  
Travelling Waves ◽  
Discrete Symmetry ◽  
Travelling Wave ◽  
Circular Cylinders ◽  
Finite Dimensional ◽  
Azimuthal Symmetry ◽  
Two Dimensional Flow ◽  
Taylor Couette ◽  
Low Dimensional ◽  
Sufficient Reduction

Results are presented from an experimental investigation of the dynamics of driven rotating flows in stadium-shaped domains. The work was motivated by questions concerning the typicality of low-dimensional dynamical phenomena which are found in Taylor-Couette flow between rotating circular cylinders. In such a system, there is continuous azimuthal symmetry and travelling-wave solutions are found. In the present study, this symmetry is broken by replacing the stationary outer circular cylinder with one which has a stadium-shaped cross-section. Thus there is now only discrete symmetry in the azimuthal direction, and travelling waves are no longer observed. To begin with, the two-dimensional flow field was investigated using numerical techniques. This was followed by an experimental study of the dynamics of flow in systems with finite vertical extent. Configurations involving both right-circular and tapered inner cylinders were considered. Dynamics were observed which correspond to known mechanisms from the theory of finite-dimensional dynamical systems. However, flow behaviour was also observed which cannot be classified in this way. Thus it is concluded that while certain low-dimensional dynamical phenomena do persist with breaking of the continuous azimuthal symmetry embodied in the Taylor-Couette system, sufficient reduction of symmetry admits behaviour at moderately low Reynolds number which is without any low-dimensional characteristics.

Flow instabilities in the wake of a rounded square cylinder

2016 ◽  
Vol 793 ◽  
pp. 915-932 ◽  
Author(s):  
Doohyun Park ◽  
Kyung-Soo Yang
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Landau Equation ◽  
Secondary Instability ◽  
Immersed Boundary ◽  
Neutral Stability ◽  
Square Cylinder ◽  
Flow Topology ◽  
Sharp Edges ◽  
Primary Instability

Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius $r$ in each edge of a square cylinder of height $d$. Nine cases of rounding were considered, ranging from a square cylinder ($r/d=0$) to a circular cylinder ($r/d=0.5$) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number (Re) and the edge-radius ratio ($r/d$). For low Re, the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ($Re_{c1}$), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ($Re_{c2}$). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon $r/d$. The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon $r/d$.

Analysis of Plug Zones in Steady Flow in Undulating Channels

2011 ◽  
Author(s):  
Mario F. Letelier ◽  
Pierre Svensson ◽  
Dennis A. Siginer ◽  
Juan S. Stockle
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Shear Stress ◽  
Steady Flow ◽  
Industrial Applications ◽  
Complex Flow ◽  
Bingham Plastic ◽  
Good Consistency ◽  
Constant Shear Stress ◽  
Flow Through

Undulating channels are important in several industrial applications related to heat transfer and also as a modeling resource for complex flow through pores or fibers. In this paper, flow of a Bingham plastic in an undulating channel is studied. An analytical model is developed which allows to determine the regions in the flow where solid (plug) or quasisolid behavior appears. The parameters considered are the dimensionless yield stress, the channel amplitude and the Reynolds number. A variety of cases are presented, in which the characteristics of the flows are described. The main results are presented by means of the streamlines, isobars, constant shear stress lines, isovelocities, and velocity profiles. Good consistency if found among all variables analyzed.

scholarly journals The effect of Reynolds number on turbulent drag reduction by streamwise travelling waves

2014 ◽  
Vol 759 ◽  
pp. 28-55 ◽  
Author(s):  
Edward Hurst ◽  
Qiang Yang ◽  
Yongmann M. Chung
Keyword(s):  
Reynolds Number ◽  
Drag Reduction ◽  
Flow Control ◽  
Travelling Waves ◽  
Reynolds Numbers ◽  
Travelling Wave ◽  
Control Parameters ◽  
Scaling Parameter ◽  
High Reynolds Numbers ◽  
High Reynolds

AbstractThis paper exploits the turbulent flow control method using streamwise travelling waves (Quadrio et al. J. Fluid Mech., vol. 627, 2009, pp. 161–178) to study the effect of Reynolds number on turbulent skin-friction drag reduction. Direct numerical simulations (DNS) of a turbulent channel flow subjected to the streamwise travelling waves of spanwise wall velocity have been performed at Reynolds numbers ranging from $\mathit{Re}_{{\it\tau}}=200$ to 1600. To the best of the authors’ knowledge, this is the highest Reynolds number attempted with DNS for this type of flow control. The present DNS results confirm that the effectiveness of drag reduction deteriorates, and the maximum drag reduction achieved by travelling waves decreases significantly as the Reynolds number increases. The intensity of both the drag reduction and drag increase is reduced with the Reynolds number. Another important finding is that the value of the optimal control parameters changes, even in wall units, when the Reynolds number is increased. This trend is observed for the wall oscillation, stationary wave, and streamwise travelling wave cases. This implies that, when the control parameters used are close to optimal values found at a lower Reynolds number, the drag reduction deteriorates rapidly with increased Reynolds number. In this study, the effect of Reynolds number for the travelling wave is quantified using a scaling in the form $\mathit{Re}_{{\it\tau}}^{-{\it\alpha}}$. No universal constant is found for the scaling parameter ${\it\alpha}$. Instead, the scaling parameter ${\it\alpha}$ has a wide range of values depending on the flow control conditions. Further Reynolds number scaling issues are discussed. Turbulent statistics are analysed to explain a weaker drag reduction observed at high Reynolds numbers. The changes in the Stokes layer and also the mean and root-mean-squared (r.m.s.) velocity with the Reynolds number are also reported. The Reynolds shear stress analysis suggests an interesting possibility of a finite drag reduction at very high Reynolds numbers.

scholarly journals Axisymmetric travelling waves in annular sliding Couette flow at finite and asymptotically large Reynolds number

2013 ◽  
Vol 720 ◽  
pp. 582-617 ◽  
Author(s):  
K. Deguchi ◽  
A. G. Walton
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Vortical Flow ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Equilibrium Solutions ◽  
Specific Flow

AbstractThe relationship between numerical finite-amplitude equilibrium solutions of the full Navier–Stokes equations and nonlinear solutions arising from a high-Reynolds-number asymptotic analysis is discussed for a Tollmien–Schlichting wave-type two-dimensional vortical flow structure. The specific flow chosen for this purpose is that which arises from the mutual axial sliding of co-axial cylinders for which nonlinear axisymmetric travelling-wave solutions have been discovered recently by Deguchi & Nagata (J. Fluid Mech., vol. 678, 2011, pp. 156–178). We continue this solution branch to a Reynolds number $R= 1{0}^{8} $ and confirm that the behaviour of its so-called lower branch solutions, which typically produce a smaller modification to the laminar state than the other solution branches, quantitatively agrees with the axisymmetric asymptotic theory developed in this paper. We further find that this asymptotic structure breaks down when the disturbance wavelength is comparable with $R$. The new structure which replaces it is investigated and the governing equations are derived and solved. The flow visualization of the resultant solutions reveals that they possess a streamwise localized structure, with the trend agreeing qualitatively with full Navier–Stokes solutions for relatively long-wavelength disturbances.

scholarly journals Self-sustaining dual critical layer states in plane Poiseuille–Couette flow at large Reynolds number

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180881 ◽  
Author(s):  
Rishi Kumar ◽  
Andrew Walton
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Three Dimensional ◽  
Phase Speed ◽  
Wave Structure ◽  
Travelling Wave ◽  
Basic Flow ◽  
Large Reynolds Number ◽  
Amplitude Dependence ◽  
Critical Layers

The nonlinear stability of plane Poiseuille–Couette flow subjected to three-dimensional disturbances is studied asymptotically at large Reynolds number R . By analysing the nature of the instability for increasing disturbance size Δ, the scaling Δ =  O ( R −1/3 ) is identified at which a strongly nonlinear neutral wave structure emerges, involving the interaction of two inviscid critical layers. The striking feature of this structure is that the travelling wave disturbances have both streamwise and spanwise wavelengths comparable to the channel width, with an associated phase speed of O (1). An alternative method to the classical balancing of phase shifts is proposed, involving vorticity jumps, that uses a global property of the flow-field and enables the amplitude-dependence of the neutral modes to be determined in terms of the wavenumbers and the properties of the basic flow. Numerical computation of the Rayleigh equation which governs the flow outside of the critical layers shows that neutral solutions exist for non-dimensional wall sliding speeds in the range 0 ≤  V  < 2. It transpires that the critical layers merge and the asymptotic structure referred to above breaks down both in the large-amplitude limit and the limit V → 2 when the maximum of the basic flow becomes located at the upper wall.

Stochastic generalised KPP equations

1996 ◽  
Vol 126 (5) ◽  
pp. 957-983 ◽  
Author(s):  
I. M. Davies ◽  
A. Truman ◽  
H. Z. Zhao
Keyword(s):  
White Noise ◽  
Numerical Solutions ◽  
Travelling Waves ◽  
Wave Structure ◽  
Travelling Wave ◽  
Wave Form ◽  
Kpp Equations ◽  
Image Position ◽  
Theoretical Results ◽  
Good Agreement

We classify multiplicative white noise perturbationsk(·)dw, of generalised KPP equations and their effects on deterministic approximate travelling wave solutions by the behaviour of, the solutions of the stochastic generalised KPP equations converge to deterministic approximate travelling waves and ifbeing an associated potential energy, Фsa solution of the corresponding classical mechanical equations of Newton,Dbeing a certain domain inR1×Rrthen the white noise perturbations essentially destroy the wave structure and force the solutions to die down.For the case(suppose the existence of the limit) we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations. Numerical solutions are included and give good agreement with theoretical results.

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