Mathematics of Large Eddy Simulation of Turbulent Flows

by Berselli, L. C.; Iliescu, T.; Layton, W. J.

• ISBN13: 9783540263166
• ISBN10: 3540263160
• Format: Hardcover
• Copyright: 2006-01-31
• Publisher: Springer Nature

Summary

This highly praised introductory treatment describes the parallels between statistical physics and finance - both those established in the 100-year long interaction between these disciplines, as well as new research results on financial markets.The random-walk technique, well known in physics, is also the basic model in finance, upon which are built, for example, the Black-Scholes theory of option pricing and hedging, plus methods of portfolio optimization. Here the underlying assumptions are assessed critically. Using empirical financial data and analogies to physical models such as fluid flows, turbulence, or superdiffusion, the book develops a more accurate description of financial markets based on random walks. With this approach, novel methods for derivative pricing and risk management can be formulated. Computer simulations of interacting-agent models provide insight into the mechanisms underlying unconventional price dynamics. It is shown that stock exchange crashes can be modelled in ways analogous to phase transitions and earthquakes, and sometimes have even been predicted successfully.This third edition of "The Statistical Mechanics of Financial Markets" especially stands apart from other treatments because it offers new chapters containing a practitioner's treatment of two important current topics in banking: the basic notions and tools of risk management and capital requirements for financial institutions, including an overview of the new Basel II capital framework which may well set the risk management standards in scores of countries for years to come.

Table of Contents

Part I Introduction
Introduction	3	(26)
Characteristics of Turbulence	6	(2)
What are Useful Averages?	8	(6)
Conventional Turbulence Models	14	(2)
Large Eddy Simulation	16	(1)
Problems with Boundaries	17	(1)
The Interior Closure Problem in LES	18	(2)
Eddy Viscosity Closure Models in LES	20	(2)
Closure Models Based on Systematic Approximation	22	(3)
Mixed Models	25	(1)
Numerical Validation and Testing in LES	26	(3)
The Navier-Stokes Equations	29	(42)
An Introduction to the NSE	29	(3)
Derivation of the NSE	32	(4)
Boundary Conditions	36	(1)
A Few Results on the Mathematics of the NSE	37	(25)
Notation and Function Spaces	38	(4)
Weak Solutions in the Sense of Leray-Hopf	42	(1)
The Energy Balance	43	(4)
Existence of Weak Solutions	47	(7)
More Regular Solutions	54	(8)
Some Remarks on the Euler Equations	62	(3)
The Stochastic Navier-Stokes Equations	65	(3)
Conclusions	68	(3)
Part II Eddy Viscosity Models
Introduction to Eddy Viscosity Models	71	(34)
Introduction	71	(1)
Eddy Viscosity Models	72	(5)
Variations on the Smagorinsky Model	77	(4)
Van Driest Damping	78	(1)
Alternate Scalings	78	(2)
Models Acting Only on the Smallest Resolved Scales	80	(1)
Germano's Dynamic Model	80	(1)
Mathematical Properties of the Smagorinsky Model	81	(21)
Further Properties of Monotone Operators	93	(9)
Backscatter and the Eddy Viscosity Models	102	(1)
Conclusions	103	(2)
Improved Eddy Viscosity Models	105	(18)
Introduction	105	(6)
The Gaussian--Laplacian Model (GL)	111	(6)
Mathematical Properties	112	(5)
k -- ε Modeling	117	(4)
Selective Models	118	(3)
Conclusions	121	(2)
Uncertainties in Eddy Viscosity Models and Improved Estimates of Turbulent Flow Functionals	123	(12)
Introduction	123	(1)
The Sensitivity Equations of Eddy Viscosity Models	124	(3)
Calculating fδ = ∂/∂δf	126	(1)
Boundary Conditions for the Sensitivities	127	(1)
Improving Estimates of Functionals of Turbulent Quantities	127	(3)
Conclusions: Are u and p Enough?	130	(5)
Part III Advanced Models
Basic Criteria for Subfilter-scale Modeling	135	(8)
Modeling the Subfilter-scale Stresses	135	(1)
Requirements for a Satisfactory Closure Model	136	(7)
Closure Based on Wavenumber Asymptotics	143	(52)
The Gradient (Taylor) LES Model	145	(9)
Derivation of the Gradient LES Model	145	(2)
Mathematical Analysis of the Gradient LES Model	147	(6)
Numerical Validation and Testing	153	(1)
The Rational LES Model (RLES)	154	(25)
Mathematical Analysis for the Rational LES Model	157	(13)
On the Breakdown of Strong Solutions	170	(7)
Numerical Validation and Testing	177	(2)
The Higher-order Subfilter-scale Model (HOSFS)	179	(14)
Derivation of the HOSFS Model	179	(2)
Mathematical Analysis of the HOSFS Model	181	(7)
Numerical Validation and Testing	188	(5)
Conclusions	193	(2)
Scale Similarity Models	195	(32)
Introduction	195	(5)
The Bardina Model	195	(5)
Other Scale Similarity Models	200	(1)
Germano Dynamic Model	200	(1)
The Filtered Bardina Model	200	(1)
The Mixed-scale Similarity Model	201	(1)
Recent Ideas in Scale Similarity Models	201	(4)
The S4 = Skew-symmetric Scale Similarity Model	205	(8)
Analysis of the Model	206	(2)
Limit Consistency and Verifiability of the S4 Model	208	(5)
The First Energy-sponge Scale Similarity Model	213	(6)
``More Accurate'' Models	217	(2)
The Higher Order, Stolz-Adams Deconvolution Models	219	(4)
The van Cittert Approximations	220	(3)
Conclusions	223	(4)
Part IV Boundary Conditions
Filtering on Bounded Domains	227	(26)
Filters with Nonconstant Radius	229	(11)
Definition of the Filtering	230	(4)
Some Estimates of the Commutation Error	234	(6)
Filters with Constant Radius	240	(11)
Derivation of the Boundary Commutation Error (BCE)	241	(5)
Estimates of the BCE	246	(3)
Error Estimates for a Weak Form of the BCE	249	(1)
Numerical Approximation of the BCE	250	(1)
Conclusions	251	(2)
Near Wall Models in LES	253	(22)
Introduction	253	(1)
Wall Laws in Conventional Turbulence Modeling	254	(2)
Current Ideas in Near Wall Modeling for LES	256	(3)
New Perspectives in Near Wall Models	259	(13)
The 1/7th Power Law in 3D	261	(5)
The 1/nth Power Law in 3D	266	(2)
A Near Wall Model for Recirculating Flows	268	(2)
A NWM for Time-fluctuating Quantities	270	(1)
A NWM for Reattachment and Separation Points	271	(1)
Conclusions	272	(3)
Part V Numerical Tests
Variational Approximation of LES Models	275	(20)
Introduction	275	(1)
LES Models and their Variational Approximation	276	(5)
Variational Formulation	277	(4)
Examples of Variational Methods	281	(1)
Spectral Methods	281	(1)
Finite Element Methods	282	(1)
Spectral Element Methods	282	(1)
Numerical Analysis of Variational Approximations	282	(3)
Introduction to Variational Multiscale Methods (VMM)	285	(4)
Eddy Viscosity Acting on Fluctuations as a VMM	289	(4)
Conclusions	293	(2)
Test Problems for LES	295	(32)
General Comments	295	(1)
Turbulent Channel Flows	296	(24)
Computational Setting	297	(1)
Definition of Reτ	298	(2)
Initial Conditions	300	(1)
Statistics	301	(2)
LES Models Tested	303	(2)
Numerical Method and Numerical Setting	305	(2)
A Posteriori Tests for Reτ = 180	307	(3)
A Posteriori Tests for Reτ = 395	310	(2)
Backscatter in the Rational LES Model	312	(3)
Numerical Results	315	(3)
Summary of Results	318	(2)
A Few Remarks on Isotropic Homogeneous Turbulence	320	(4)
Computational Setting	321	(1)
Initial Conditions	322	(1)
Experimental Results	323	(1)
Computational Cost	323	(1)
LES of the Comte-Bellot Corrsin Experiment	324	(1)
Final Remarks	324	(3)
References	327	(18)
Index	345