scholarly journals Generalizing the Multimodal Method for the Levitating Drop Dynamics

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
M. O. Chernova ◽  
I. A. Lukovsky ◽  
A. N. Timokha
Keyword(s):  
Free Surface ◽  
Almost Periodic ◽  
Modal System ◽  
Periodic Oscillations ◽  
Weakly Nonlinear ◽  
Modal Theory ◽  
Drop Dynamics ◽  
Finite Dimensional ◽  
Multimodal Method ◽  
Good Agreement

The present paper extends the multimodal method, which is well known for liquid sloshing problems, to the free-surface problem modeling the levitating drop dynamics. The generalized Lukovsky-Miles modal equations are derived. Based on these equations an approximate modal theory is constructed to describe weakly-nonlinear axisymmetric drop motions. Whereas the drop performs almost-periodic oscillations with the frequency close to the lowest natural frequency, the theory takes a finite-dimensional form. Periodic solutions of the corresponding finite-dimensional modal system are compared with experimental and numerical results obtained by other authors. A good agreement is shown.

Nonlinear Sloshing in a Spherical Tank

2013 ◽  
Author(s):  
Odd M. Faltinsen ◽  
Alexander N. Timokha
Keyword(s):  
Steady State ◽  
Natural Frequency ◽  
Secondary Resonance ◽  
Weakly Nonlinear ◽  
Modal Theory ◽  
Spherical Tank ◽  
Theoretical Results ◽  
Multimodal Method ◽  
Comparison With Experiments ◽  
Good Agreement

Steady-state resonant sloshing in a spherical rigid tank due to horizontal harmonic excitations at the lowest natural frequency is classified by combining the nonlinear multimodal method with the Moiseev-Narimanov asymptotics. The theoretical results are validated by comparison with experiments of Sumner & Stofan (1963) and other already published model tests. A good agreement is found for the depth-to-tank radius ratios 0.2 ≤ h ≲ 1 but, when 1 ≲ h ≲ 2, secondary resonance and splashing limits the applicability of the constructed weakly-nonlinear modal theory.

Multimodal analysis of weakly nonlinear sloshing in a spherical tank

2013 ◽  
Vol 719 ◽  
pp. 129-164 ◽  
Author(s):  
Odd M. Faltinsen ◽  
Alexander N. Timokha
Keyword(s):  
Technical Note ◽  
Modal System ◽  
Weakly Nonlinear ◽  
Multimodal Analysis ◽  
Spherical Tank ◽  
Sloshing Frequency ◽  
The Stability ◽  
Asymptotic Time ◽  
Time Periodic ◽  
Good Agreement

AbstractSloshing in a spherical tank due to horizontal excitation is studied by using the nonlinear multimodal method which involves the analytically approximate sloshing modes by Faltinsen & Timokha (J. Fluid Mech., vol. 703, 2012, pp. 391–401). General fully and weakly nonlinear modal equations are derived but an emphasis is on the Moiseev–Narimanov asymptotic modal system which implies that the forcing frequency is close to the lowest natural sloshing frequency and there are no secondary resonances in the forcing frequency range leading to a nonlinear resonant amplification of double and triple harmonics in higher modes. The Moiseev–Narimanov modal system is used to construct an asymptotic time-periodic solution and, thereby, classify the corresponding steady-state wave regimes appearing as stable and unstable planar waves and swirling. The results on the stability boundaries are compared with experiments by Sumner & Stofan (1963,Tech. Rep.TN D-1991, NASA Technical Note) and Sumner (1966,Tech. Rep.TN D-3210, NASA). A good agreement is established for$0. 2\leq h\lesssim 1$. Discrepancy for higher liquid depths$1\lesssim h\lt 2$are explained by secondary resonance.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth

2002 ◽  
Vol 470 ◽  
pp. 319-357 ◽  
Author(s):  
ODD M. FALTINSEN ◽  
ALEXANDER N. TIMOKHA
Keyword(s):  
Finite Depth ◽  
Modal System ◽  
Full Account ◽  
Inviscid Flows ◽  
State Response ◽  
Linear Damping ◽  
Convergence Problems ◽  
Run Up ◽  
Fourth Order Polynomial ◽  
Good Agreement

The modal system describing nonlinear sloshing with inviscid flows in a rectangular rigid tank is revised to match both shallow fluid and secondary (internal) resonance asymptotics. The main goal is to examine nonlinear resonant waves for intermediate depth/breadth ratio 0.1 [lsim ] h/l [lsim ] 0.24 forced by surge/pitch excitation with frequency in the vicinity of the lowest natural frequency. The revised modal equations take full account of nonlinearities up to fourth-order polynomial terms in generalized coordinates and h/l and may be treated as a modal Boussinesq-type theory. The system is truncated with a high number of modes and shows good agreement with experimental data by Rognebakke (1998) for transient motions, where previous finite depth modal theories failed. However, difficulties may occur when experiments show significant energy dissipation associated with run-up at the walls and wave breaking. After reviewing published results on damping rates for lower and higher modes, the linear damping terms due to the linear laminar boundary layer near the tank's surface and viscosity in the fluid bulk are incorporated. This improves the simulation of transient motions. The steady-state response agrees well with experiments by Chester & Bones (1968) for shallow water, and Abramson et al. (1974), Olsen & Johnsen (1975) for intermediate fluid depths. When h/l [lsim ] 0.05, convergence problems associated with increasing the dimension of the modal system are reported.

Defect Structure of MEV Si Implantation in GaAs

1988 ◽  
Vol 126 ◽  
Author(s):  
S.-Tong Lee ◽  
G. Braunstein ◽  
Samuel Chen
Keyword(s):  
Free Surface ◽  
Defect Structure ◽  
Room Temperature ◽  
Surface Region ◽  
Peak Position ◽  
Lattice Disorder ◽  
Depth Profiles ◽  
Trim Calculation ◽  
Good Agreement

ABSTRACTThe defect and atomic profiles for MeV implantation of Si in GaAs were investigated using He++ channeling, TEM, and SIMS. Doses of 1–10 × 1015Si/cm2 at 1–3 MeV were used. MeV implantation at room temperature rendered only a small amount of lattice disorder in GaAs. Upon annealing at 400°C for 1 h or 800°C for 30 a, we observed a ‘defect-free’ surface region (- 1 μ for 3 MeV implant). Below this region, extensive secondary defects were formed in a band which was 0.7 μ wide and centered at 2 μ for 3 MeV implant. These defects were mostly dislocations lying in the [111] plane. SIMS depth profiles of Si implants showed the Si peak to be very close to the peak position of the defects. The experimental profiles of Si were compared to the TRIM calculation; generally good agreement existed among the peak positions.

On the evolution of a nonlinear Alfvén pulse

1999 ◽  
Vol 62 (2) ◽  
pp. 219-232 ◽  
Author(s):  
E. VERWICHTE ◽  
V. M. NAKARIAKOV ◽  
A. W. LONGBOTTOM
Keyword(s):  
Temporal Evolution ◽  
Transverse Velocity ◽  
Longitudinal Velocity ◽  
Initial Pulse ◽  
Weakly Nonlinear ◽  
Starting Position ◽  
Linearly Polarized ◽  
Nonlinear Plane ◽  
The Magnetic Field ◽  
Good Agreement

The temporal evolution of weakly nonlinear, plane, linearly polarized Alfvén pulses in a cold homogeneous plasma is investigated. A static initial pulse-like disturbance in transverse velocity produces two Alfvén pulses that travel in opposite directions along the magnetic field. The ponderomotive force of the two pulses produces a static shock in longitudinal velocity at the starting position. The travelling pulses form a shock front that is governed by the scalar Cohen–Kulsrud equation. We find good agreement between the analytical solutions we derive and the results from a fully nonlinear numerical MHD code.

A Two-Dimensional Potential Flow Model of the Wave Field Generated by a Semisubmerged Body in Heaving Motion

1988 ◽  
Vol 32 (02) ◽  
pp. 83-91
Author(s):  
X. M. Wang ◽  
M. L. Spaulding
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Potential Flow ◽  
Flow Model ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Potential Flow Model ◽  
Good Agreement

A two-dimensional potential flow model is formulated to predict the wave field and forces generated by a sere!submerged body in forced heaving motion. The potential flow problem is solved on a boundary fitted coordinate system that deforms in response to the motion of the free surface and the heaving body. The full nonlinear kinematic and dynamic boundary conditions are used at the free surface. The governing equations and associated boundary conditions are solved by a second-order finite-difference technique based on the modified Euler method for the time domain and a successive overrelaxation (SOR) procedure for the spatial domain. A series of sensitivity studies of grid size and resolution, time step, free surface and body grid redistribution schemes, convergence criteria, and free surface body boundary condition specification was performed to investigate the computational characteristics of the model. The model was applied to predict the forces generated by the forced oscillation of a U-shaped cylinder. Numerical model predictions are generally in good agreement with the available second-order theories for the first-order pressure and force coefficients, but clearly show that the third-order terms are larger than the second-order terms when nonlinearity becomes important in the dimensionless frequency range 1≤ Fr≤ 2. The model results are in good agreement with the available experimental data and confirm the importance of the third order terms.

ASYMPTOTICAL ANALYSIS OF ONE DIMENSIONAL GAS DYNAMICS EQUATIONS

2001 ◽  
Vol 6 (1) ◽  
pp. 117-128 ◽  
Author(s):  
A. Krylovas ◽  
R. Čiegis
Keyword(s):  
Gas Dynamics ◽  
Numerical Experiments ◽  
Method Of Averaging ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Resonance Case ◽  
Gas Dynamics Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Asymptotical Analysis

A method of averaging is developed for constructing a uniformly valid asymptotic solution for weakly nonlinear one dimensional gas dynamics systems. Using this method we give the averaged system, which disintegrates into independent equations for the non‐resonance systems. Conditions of the resonance for periodic and almost periodic solutions are presented. In the resonance case the averaged system is solved numerically. Some results of numerical experiments are given.

Numerical Simulations of Ship Bow and Shoulder Wave Breaking in Different Advancing Speeds

2018 ◽  
Author(s):  
Zhen Ren ◽  
Jianhua Wang ◽  
Decheng Wan
Keyword(s):  
Free Surface ◽  
Numerical Simulations ◽  
Wave Breaking ◽  
Wave Pattern ◽  
Breaking Wave ◽  
Wake Field ◽  
Bow Wave ◽  
Calm Water ◽  
Wave Phenomena ◽  
Good Agreement

The KCS model is employed for the numerical simulations to investigate the wave breaking phenomena of the bow and shoulder wave. RANS approach coupled with high resolution VOF technique is used to resolve the free surface. In order to study the speed effects on the phenomena of ship wave breaking, four different speeds, i.e. Fr = 0.26, 0.30, 0.32, 0.35, are investigated in calm water. Predicted resistance and wave patterns under Fr = 0.26 are validated with the available experiment data, and good agreement is achieved. For the Fr = 0.26 case, the wave pattern is steady, and the alternate variation of vorticity appear near the free surface is associated with the wake field. The breaking wave phenomena can be observed when the Froude number is over 0.32 and the Fr = 0.35 case shows most violent breaking bow wave. For the Fr = 0.35 case, the process of overturning and breaking of bow wave is observed clearly, and at the tail of bow wave, some breaking features of free surface are also captured. The reconnection of the initial plunger with the free surface results in a pair of counter-rotating vortex that is responsible for the second plunger and scar.

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