Computational Analysis of an Imploding Gas:
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1 1/ 31 Direct Numerical Simulation of Navier-Stokes Equations Stephen Voelkel University of Notre Dame October 19, 2011
2 2/ 31 Acknowledges Christopher M. Romick, Ph.D. Student, U. Notre Dame Dr. Joseph M. Powers, U. Notre Dame Dr. Jarek Nabrzyski, U. Notre Dame Zack Zikoski, Ph.D. Student, U. Notre Dame National Science Foundation
3 3/ 31 Outline I 1 Motivation 2 Model Equations 3 Computational Method 4 Verification 5 Implosions 6 Convergence 7 Concluding Remarks
4 4/ 31 Motivation Computational Fluid Analysis Motivating Factors Sonoluminescence caused by acoustic implosions Bubble cavitation and subsequent damage to machinery Detonation-aided implosion and higher post-detonation temperatures and pressures (Jiang, et. al. 2007)
5 5/ 31 Motivation Advanced Algorithms in Computational Physics Why More Efficient Algorithms are Important Moore s law (1965) predicts doubling of computational speed every two years. Argued by Fast, et. al. (JCP 2006), that even more improvement in speed has come from improved algorithms. In order to increase spatial and temporal domain restrictions for direct numerical simulation, DNS, we need more efficient algorithms. Manley, et. al. (Phys. Today 2008)
6 6/ 31 Model Equations Summary of Continuum Axioms In Differential Form Conservation of Mass Conservation of Linear Momenta t ρu i t Conservation of Energy ( ρ ρ t + ρu i x i = 0 + x j (ρu i u j ) = p x i + τ ij x j (e + 12 u ju j )) + x i q i x i (ρu i (e + 12 )) u ju j = x i (pu i ) + x i (τ ij u j )
7 7/ 31 Model Equations Constitutive Relations Fourier s Law q i = k T x i Newtonian Fluid with Stokes Assumption τ ij = µ ( ui x j + u j x i Thermal Equation of State: Ideal Gas Law p(ρ, T ) = ρrt ) 2 3 µ u k x k δ ij Caloric Equation of State: Calorically Perfect Gas e(t ) = e o + c v (T T o )
8 8/ 31 Model Equations An Approach to Solving the Navier-Stokes Equations Euler s equations Neglect viscous terms Small scale phenomena ignored Use shock-fitting or shock-capturing methods Direct Numerical Simulation, DNS Predicts solution of full Navier-Stokes equations No turbulence modeling No shock-fitting; shock-capturing is physically based Resolves all relevant scales (e.g. scales above the mean free path of the fluid)
9 9/ 31 Computational Method Wavelet Adaptive Mesh Refinement, WAMR Navier-Stokes equations spatially discretized as dq i dt = i (f i (q i )) (1) where i is any consistent spatial discretization (e.g. finite difference). Adaptive spatial grid determined using a wavelet transformation (Zikoski 2011) Solved with explicit time-advancement Parallel processing using message passing interface, MPI Advanced chemical kinetics and transport properties via Chemkin and Transport libraries
10 10/ 31 Computational Method Adaptive Mesh Refinement, AMR In a domain with spatial variation in property gradients, AMR can significantly reduce the number of necessary grid points. Tolerance controlled by a user-defined parameter, ɛ. e.g points define the grid, whereas if uniform, points would be needed.
11 11/ 31 Verification Verification - 1D Shock Tube Assumptions Compare the viscous solution predicted with WAMR to an inviscid, analytical solution. To achieve closed-form, analytical solution, assume: One-Dimensional Flow Negligible Viscosity Negligible Diffusive Heat Transfer Argon gas P low Rarefaction Contact Shock P high =
12 3 12/ 31 Verification Verification - 1D Shock Tube Comparison to WAMR Density ( kg / m ) Position ( cm ) Pressure ( kpa ) Position ( cm ) Temperature ( K ) Velocity ( m/s ) Position ( cm ) Position ( cm )
13 13/ 31 Verification Verification - 1D Shock Tube Comparison to WAMR Continued - r t Diagram
14 14/ 31 Verification Verification - 1D Steady Viscous Shock Structure Assumptions Assume: One-Dimensional Flow Steady State Constant Viscosity Constant Thermal Conductivity O 2 gas d x (ρu) = 0 ( d ρu 2 + p 4 ) x 3 µdu = 0 dx ( ( d ρu e + 1 x 2 u2 + p ) k dt ρ dx 4 ) 3 µudu = 0 dx
15 15/ 31 Verification Verification - 1D Steady Viscous Shock Structure Two States Integrating yields ρu = ρ o u o ρu 2 + p 4 3 µdu dx = ρ ou 2 o + p o ( ρu e u2 + p ) k dt ρ dx 4 ( 3 µudu dx = ρ ou o e o u2 o + p ) o ρ o where p = ρrt e e o = c v (T T o ) and ρ o, p o, u o, and e o represent the initial state far from the shock.
16 16/ 31 Verification Verification - 1D Steady Viscous Shock Structure Coupled ODE s Solving for du/dx and dt/dx results in two coupled, nonlinear ordinary differential equations: du dx = 3 [ ( ρ o u o T (u u o ) + R 4 µ u T )] o u o dt dx = ρ ou o k [ c v (T T o ) 1 2 (u u o) 2 + RT o ( u u o 1 )]
17 17/ 31 Verification Verification - 1D Steady Viscous Shock Structure Equilibrium States Setting du/dx and dt/dx equal to zero gives two equilibrium conditions: T 1 = T o u 1 = u o and T 2 = u 2 = ( ) ( ) 2γ (γ 1) T 2 o 1 + γ (γ 6) (γ + 1) 2 u 2 + o (γ + 1) 2 T o ( ) 2 (γ 1) R (γ + 1) 2 u 2 o ( ) ( ) 2Rγ To γ 1 + u o γ + 1 u o γ + 1
18 18/ 31 Verification Verification - 1D Steady Viscous Shock Structure Phase Plot Two equilibria Shock frame of reference Source becomes a sink Travels backwards from the shocked state to the initial, unshocked state Unshocked State T ( K ) Shocked State u ( m/s )
19 Verification Verification - 1D Steady Viscous Shock Structure Comparison to WAMR Initial Conditions M s = 1.61 T o = 300 K u o = 0 m/s P o = 12.0 kpa Shocked Conditions T 2 = 383 K u 2 = 225 m/s P 2 = 34.6 kpa Mean free path, λ = 300 nm T (K) Error Analytical Solution WAMR Prediction x/λ x/λ 19/ 31
20 20/ 31 Implosions Implosion Problem Description Initially, a high and low pressure state are separated by an infinitesimally thin, impermeable diaphram Diaphram is burst at t = 0 s The high and low pressure states interact, producing shock waves, contact discontinuities, and rarefaction waves. Compression waves implode towards a focal point y High Pressure State Low Pressure State Diaphram x
21 21/ 31 Implosions Implosion Problem Parameters Gas: Argon Domain: x [ 50, 50] µm and y [ 50, 50] µm Focal point located at (x, y) = (0, 0). Uniform temperature of 300 K P high = 4 atm P low = 1 atm
22 22/ 31 Implosions Circular Diaphram!"#$#%&# '"#($#%&# 1 Simple diaphram 2 Implosion reaches focal point Rapid ascent to peak T and p Shock structures are readily apparent 3 Shock rebounds across contact discontinuity 4 Symmetry breaks when rarefaction wave hits boundary )"#*$#%&# ("#!'$#%&#
23 23/ 31 Implosions r t Analysis
24 24/ 31 Implosions Octagonal Diaphram 1 Complex diaphram (0 ns) 2 Implosion reaches focal point Structures fold in on one another Beginning of pattern formation 3 Shock rebounds across contact discontinuity, creating complex, symmetrical pattern 4 Patterns continue to form as shock continues to rebound outward!"#$#%&# )"#*$#%&# '"#($#%&# ("#!'$#%&#
25 25/ 31 Implosions Octagonal Diaphram Video Video shows magnitude of gradient of magnitude of velocity vector, i.e. ( ) u 2 ( ) u 2 u = + (2) x y
26 26/ 31 Implosions Diaphrams with Varying Levels of Symmetry Consider four diaphrams with differing levels of symmetry:
27 27/ 31 Implosions Focal Point Temperature Analysis The temperature at the focal point ranges from six to eight times greater than the initial temperature, T o. Peak temperature obtained at the moment of full shock implosion T c/t o T = 2,500 K Circle Octagon Triangle Asymmetric t/ t i
28 28/ 31 Implosions Grid Analysis Number of collocation points used by WAMR versus time Number of points reaches a minimum as the shock implodes Number of points dramatically increases as the reflected shock travels through the contact discontinuity
29 29/ 31 Convergence Convergence of Structures As the tolerance, ɛ, approaches zero, there is a convergence to a structure. t = 60 ns.!"#"$"%"&' ()"!"#"&"%"&' ()"!"#"&"%"&' (*"!"#"&"%"&' (+"
30 30/ 31 Convergence Error and Tolerance Consider the lowest tolerance as the exact solution (ɛ = ) Error defined as the maximum difference between the exact solution and the approximate solution, ( ) ρapp ρ exact E = max ρ exact Decrease in error shows a convergence to a solution E ɛ Tolerance,
31 31/ 31 Concluding Remarks Concluding Remarks WAMR allows for DNS calculations WAMR verified with 1D shock tube and steady viscous shock structure Implosion of Argon shows interaction between imploding compression waves, contact discontinuities, and rarefaction waves. Dramatic increase in temperature as compression waves implode toward a focal point Complex pattern formation in octagonal diaphram Convergence of detailed structures
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