scholarly journals Nonlinear control of unsteady finite-amplitude perturbations in the Blasius boundary-layer flow

2013 ◽  
Vol 737 ◽  
pp. 440-465 ◽  
Author(s):  
S. Cherubini ◽  
J.-C. Robinet ◽  
P. De Palma
Keyword(s):  
Stokes Equations ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Control Procedure ◽  
Normal Velocity ◽  
Optimal Control Strategy ◽  
Navier Stokes Equations ◽  
Target Time ◽  
Layer Flow ◽  
Laminar State

AbstractThe present work provides an optimal control strategy, based on the nonlinear Navier–Stokes equations, aimed at hampering the rapid growth of unsteady finite-amplitude perturbations in a Blasius boundary-layer flow. A variational procedure is used to find the blowing and suction control law at the wall providing the maximum damping of the energy of a given perturbation at a given target time, with the final aim of leading the flow back to the laminar state. Two optimally growing finite-amplitude initial perturbations capable of leading very rapidly to transition have been used to initialize the flow. The nonlinear control procedure has been found able to drive such perturbations back to the laminar state, provided that the target time of the minimization and the region in which the blowing and suction is applied have been suitably chosen. On the other hand, an equivalent control procedure based on the linearized Navier–Stokes equations has been found much less effective, being not able to lead the flow to the laminar state when finite-amplitude disturbances are considered. Regions of strong sensitivity to blowing and suction have been also identified for the given initial perturbations: when the control is actuated in such regions, laminarization is also observed for a shorter extent of the actuation region. The nonlinear optimal blowing and suction law consists of alternating wall-normal velocity perturbations, which appear to modify the core flow structures by means of two distinct mechanisms: (i) a wall-normal velocity compensation at small times; (ii) a rotation-counterbalancing effect al larger times. Similar control laws have been observed for different target times, values of the cost parameter, and streamwise extents of the blowing and suction zone, meaning that these two mechanisms are robust features of the optimal control strategy, provided that the nonlinear effects are taken into account.

Numerical Prediction of Impact-Related Wave Loads on Ships

2006 ◽  
Vol 129 (1) ◽  
pp. 39-47 ◽  
Author(s):  
Thomas E. Schellin ◽  
Ould el Moctar
Keyword(s):  
Stokes Equations ◽  
Numerical Procedure ◽  
Navier Stokes ◽  
Ship Motions ◽  
Normal Velocity ◽  
Wave Loads ◽  
Navier Stokes Equations ◽  
Critical Areas ◽  
Strip Theory ◽  
A Chain

We present a numerical procedure to predict impact-related wave-induced (slamming) loads on ships. The procedure was applied to predict slamming loads on two ships that feature a flared bow with a pronounced bulb, hull shapes typical of modern offshore supply vessels. The procedure used a chain of seakeeping codes. First, a linear Green function panel code computed ship responses in unit amplitude regular waves. Ship speed, wave frequency, and wave heading were systematically varied to cover all possible combinations likely to cause slamming. Regular design waves were selected on the basis of maximum magnitudes of relative normal velocity between ship critical areas and wave, averaged over the critical areas. Second, a nonlinear strip theory seakeeping code determined ship motions under design wave conditions, thereby accounting for the nonlinear pressure distribution up to the wave contour and the frequency dependence of the radiation forces (memory effect). Third, these nonlinearly computed ship motions constituted part of the input for a Reynolds-averaged Navier–Stokes equations code that was used to obtain slamming loads. Favorable comparison with available model test data validated the procedure and demonstrated its capability to predict slamming loads suitable for design of ship structures.

Numerical Prediction of Impact-Related Wave Loads on Ships

2006 ◽  
Author(s):  
Thomas E. Schellin ◽  
Ould El Moctar
Keyword(s):  
Stokes Equations ◽  
Numerical Procedure ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Ship Motions ◽  
Normal Velocity ◽  
Wave Loads ◽  
Critical Areas ◽  
Strip Theory ◽  
A Chain

We present a numerical procedure to predict impact-related wave-induced (slamming) loads on ships. The procedure was applied to predict slamming loads on two ships that feature a flared bow with a pronounced bulb, hull shapes typical of modern offshore supply vessels. The procedure used a chain of seakeeping codes. First, a linear Green function panel code computed ship responses in unit amplitude regular waves. Wave frequency and wave heading were systematically varied to cover all possible combinations likely to cause slamming. Regular design waves were selected on the basis of maximum magnitudes of relative normal velocity between ship critical areas and wave, averaged over the critical areas. Second, a nonlinear strip theory seakeeping code determined ship motions under design wave conditions, thereby accounting for the ship’s forward speed, the swell-up of water in finite amplitude waves, as well as the ship’s wake that influences the wave elevation around the ship. Third, these nonlinearly computed ship motions constituted part of the input for a Reynolds-averaged Navier-Stokes equations (RANSE) code that was used to obtain slamming loads. Favourable comparison with available model test data validated the procedure and demonstrated its capability to predict slamming loads suitable for design of ship structures.

scholarly journals Subcritical transition in channel flows

2002 ◽  
Vol 451 ◽  
pp. 35-97 ◽  
Author(s):  
S. JONATHAN CHAPMAN
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Domain Of Attraction ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations ◽  
Laminar State

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as Rγ say, with γ < 0), so that small but finite perturbations lead to transition. Trefethen et al. (1993) conjectured that in fact γ <−1. Subsequent numerical experiments by Lundbladh, Henningson & Reddy (1994) indicated that for streamwise initial perturbations γ =−1 and −7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations γ =−5/4 and −7/4 Here, through a formal asymptotic analysis of the Navier–Stokes equations, it is found that for streamwise initial perturbations γ =−1 and −3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations γ =−1 and −5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.

A mechanism for bypass transition from localized disturbances in wall-bounded shear flows

1993 ◽  
Vol 250 ◽  
pp. 169-207 ◽  
Author(s):  
Dan S. Henningson ◽  
Anders Lundbladh ◽  
Arne V. Johansson
Keyword(s):  
Shear Flows ◽  
Stokes Equations ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Normal Velocity ◽  
Bypass Transition ◽  
Nonlinear Interactions ◽  
Streamwise Vorticity ◽  
Low Velocity ◽  
Transfer Energy

The linear, nonlinear and breakdown stages in the transition of localized disturbances in plane Poiseuille flow is studied by direct numerical simulations and analysis of the linearized Navier–Stokes equations. Three-dimensionality plays a key role and allows for algebraic growth of the normal vorticity through the linear lift-up mechanism. This growth primarily generates elongated structures in the streamwise direction since it is largest at low streamwise wavenumbers. For finite-amplitude disturbances such structures will be generated essentially independent of the details of the initial disturbance, since the preferred nonlinear interactions transfer energy to low streamwise wavenumbers. The nonlinear interactions also give a decrease in the spanwise scales. For the stronger initial disturbances the streamwise vorticity associated with the slightly inclined streaks was found to roll up into distinct streamwise vortices in the vicinity of which breakdown occurred. The breakdown starts with a local rapid growth of the normal velocity bringing low-speed fluid out from the wall. This phenomenon is similar to the low-velocity spikes previously observed in transition experiments. Soon thereafter a small turbulent spot is formed. This scenario represents a bypass of the regular Tollmien–Schlichting, secondary instability process. The simulations have been carried out with a sufficient spatial resolution to ensure an accurate description of all stages of the breakdown and spot formation processes. The generality of the observed processes is substantiated by use of different types of initial disturbances and by Blasius boundary-layer simulations. The present results point in the direction of universality of the observed transition mechanisms for localized disturbances in wall-bounded shear flows.

scholarly journals Derivation of a Viscous Serre–Green–Naghdi Equation: An Impasse?

Fluids ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 135
Author(s):  
Denys Dutykh ◽  
Hervé V.J. Le Meur
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Euler Equations ◽  
Stokes Equations ◽  
Current Status ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Bottom Region ◽  
Flow Domain ◽  
Layer Flow

In this article, we present the current status of the derivation of a viscous Serre–Green–Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary layer) is described by Navier–Stokes equations. We consider a particular regime binding the Reynolds number and the shallowness parameter. The computations presented in this article are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prandtl-like equation that we claim to be irreducible. Further approximations are necessary to obtain a tractable model.

scholarly journals Stokes’s Fundamental Contributions to Fluid Dynamics

2019 ◽  
pp. 115-128
Author(s):  
Peter Lynch
Keyword(s):  
Fluid Dynamics ◽  
Water Waves ◽  
Stokes Equations ◽  
Weather Prediction ◽  
Finite Amplitude ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Basic Equations ◽  
Primary Focus

George Gabriel Stokes made fundamental mathematical contributions to fluid dynamics that had profound practical consequences. The basic equations formulated by him play a central role in numerical weather prediction, in the simulation of blood flow in the body and in countless other important applications. In this chapter the primary focus is on the two most important areas of Stokes’s work on fluid dynamics, the derivation of the Navier–Stokes equations and the theory of finite amplitude oscillatory water waves.

scholarly journals Transient investigation of stack-driven airflow process through rectangular cross-ventilated building with two vents in the absence of opposing flow in the upper opening

2018 ◽  
Vol 7 (3) ◽  
pp. 1249
Author(s):  
A L. Muhammad ◽  
M Z. Ringim ◽  
L A. Isma’il
Keyword(s):  
Natural Ventilation ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Temperature Profiles ◽  
Force Term ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
New Class ◽  
Layer Flow ◽  
Separation Of Variable

Natural ventilation of building provides improvement of internal comfort and air quality conditions leading to a significant reduction of cool-ing energy consumption. Design of natural ventilation systems for many types of building is based on buoyancy forces. However, external wind flow can have significant effects on buoyancy- driven natural ventilation. The paper was concerned with transient investigation of airflow through two vents in the absence of opposing flow in the upper opening. A flow of this type represents a new class of boundary- layer flow problems in the building. Moreover, this is an exact solution of the complete Navier- Stokes Equations (including, buoyancy force term), which were then Non Dimensionalised using some dimensionless parameters and then solved analytically by separation of variable methods in which, the behavior of parameters in the results were predicts the velocity, temperature profiles together with volumetric airflow and mass transfer. The results were then evaluated numerically for several sets of values of the parameters in order to ascertain the best for optimal ventilation.  

scholarly journals SHOALING OF FINITE-AMPLITUDE WAVES ON PLANE BEACHES

1970 ◽  
Vol 1 (12) ◽  
pp. 21
Author(s):  
Robert K-C Chan ◽  
Robert L. Street
Keyword(s):  
Water Waves ◽  
Stokes Equations ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Finite Difference Equations ◽  
Rectangular Mesh ◽  
Finite Amplitude Waves ◽  
Large Water ◽  
Marker And Cell Method

This work focuses on the shoaling of large water waves with particular application to storm-generated waves and tsunamis The specific objective is the exact simulation on a digital computer of finite-amplitude waves advancing on a beach of constant slope The study is based on the simulation technique called SUMMAC (the Stanford-University-Modified Marker-And-Cell Method) The flow field is represented by a rectangular mesh of cells and by a line of hypothetical particles which defines the free surface Based on the Navier-Stokes equations, finite-difference equations were derived so that the entire flow configuration could be advanced through a finite increment of time The pressure and velocity components are used directly as the dependent variables Through extensive analyses and numerical experiments, this scheme was found to be computationally stable if the cell size and the time increment are properly selected As a specific example, the dynamics of a solitary wave passing from a zone of constant depth onto a sloping beach were simulated Primary attention was focused on the details of the water particle motions and the changes in the amplitude and shape of the wave as it climbed the slope The computed results are compared with the experiments with good agreement.

A Numerical Study of Vortex Shedding From a Square Cylinder With Ground Effect

1997 ◽  
Vol 119 (3) ◽  
pp. 512-518 ◽  
Author(s):  
Robert R. Hwang ◽  
Chia-Chi Yao
Keyword(s):  
Vortex Shedding ◽  
Stokes Equations ◽  
Numerical Study ◽  
Subsequent Development ◽  
Ground Effect ◽  
Navier Stokes ◽  
Square Cylinder ◽  
Navier Stokes Equations ◽  
Layer Flow ◽  
Volume Method

A numerical study has been conducted to investigate the behavior of the vortical wake created by a square cylinder placed in a laminar boundary-layer flow. The calculations are performed by solving the unsteady 2D Navier-Stokes equations with a finite-volume method. The Reynolds-number regime investigated is from 500 to 1500. Another parameter that is varied is the distance of the cylinder from the wall. The initial and subsequent development of the vortex shedding phenomenon are investigated. The presence of the wall is found to have strong effects on the properties of these vortices, as well as lift, drag, and Strouhal number.

Increasing Adiabatic Film-Cooling Effectiveness by Using an Upstream Ramp

2006 ◽  
Vol 129 (4) ◽  
pp. 464-471 ◽  
Author(s):  
Sangkwon Na ◽  
Tom I-P. Shih
Keyword(s):  
Film Cooling ◽  
Stokes Equations ◽  
Lateral Spreading ◽  
Navier Stokes ◽  
Backward Facing Step ◽  
Navier Stokes Equations ◽  
Film Cooling Effectiveness ◽  
Adiabatic Effectiveness ◽  
Layer Flow ◽  
Film Cooling Holes

A new design concept is presented to increase the adiabatic effectiveness of film cooling from a row of film-cooling holes. Instead of shaping the geometry of each hole; placing tabs, struts, or vortex generators in each hole; or creating a trench about a row of holes, this study proposes a geometry modification upstream of the holes to modify the approaching boundary-layer flow and its interaction with the film-cooling jets. Computations, based on the ensemble-averaged Navier–Stokes equations closed by the realizable k‐ε turbulence model, were used to examine the usefulness of making the surface just upstream of a row of film-cooling holes into a ramp with a backward-facing step. The effects of the following parameters were investigated: angle of the ramp (8.5deg, 10deg, 14deg), distance between the backward-facing step and the row of film-cooling holes (0.5D,D), blowing ratio (0.36, 0.49, 0.56, 0.98), and “sharpness” of the ramp at the corners. Results obtained show that an upstream ramp with a backward-facing step can greatly increase surface adiabatic effectiveness. The laterally averaged adiabatic effectiveness with a ramp can be two or more times higher than without the ramp by increasing upstream and lateral spreading of the coolant.

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