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Computational Fluid Dynamics
        Modern Applications, Challenges and Future
                         Trends

             Arab Academy for Science, Technology and Maritime Transport, 28th March 2012




                                   Dr. Khalid M. Saqr
                           B.Sc (Alex. Uni.), M.Eng (UTM), PhD (UTM)


                                   © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
28/03/2012
Outlines
I.         CFD in a nutshell
       •     Tool or an engineering science?
       •     Frame of reference of CFD
II.        Modern CFD codes
       •     Building blocks of modern codes
       •     Commercial vs. open source codes
III.       Modern applications of CFD
       •     Industrial sector
       •     Medicine and biomechanics
       •     Homeland security
IV.        CFD for combustion applications
       •     Modeling of turbulent vortex flames
V.         Contemporary challenges
       •     Physical models
       •     Computational requirements
       •     Benchmarking and validation
VI.        Future trends
       •     Mesh-free methods
       •     Low-cost HPC
       •     Runtime validation concept


                    © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
I.          CFD in a nutshell                                                          (definition)




                                                      ?
CFD: An engineering tool…? Or an independent branch of engineering
science ?
                                                                                       Computational
                                                                                       homodynamic



                                                    Computational
                                                    Oceanography


     2nd World War weaponry
         1930s – 1940s                                              2000 – 2010s
                 © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
I.        CFD in a nutshell                                                                  (definition)

                                                                    Computer Science
            Fluid Mechanics
                                                                    &
                                                                    Software Engineering
      Heat Transfer


                                             CFD
  Electromagnetic field                                                                       Processor development


        Chemical reaction                                                                   Parallel computing


                                           Physical
                                           & Life Sciences
            Engineering science


                          Geophysics                             Vascular medicine

                                                   Oceanography
21st Century definition of CFD:
Solving the governing equations of fluid flow -by using computational methods-
under various physical conditions (e.g. heat transfer, radiation, electromagnetic
field, nuclear reaction…etc.) to study different phenomena encountered in the
numerous branches of physical science.

                      © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
I.       CFD in a nutshell                                                              (frame of reference)

                Frame of reference of engineering CFD

Theories of        Continuum                     Molecular /                        Quantum theory
matter:            hypothesis                    Kinetic theory


Mechanics:        Newtonian                      Relativistic



Chaos theory:     Deterministic                  Random chaos
                  chaos


1. Matter (i.e. fluid) is assumed to be a continuum, and its properties
and characteristics can be described by field functions.
2. These field functions are functions in Euclidian space and time, and
their frame of reference is an inertial one.
3. The sensitive dependence on initial conditions described by the
governing partial differential equation does not involve any random
events


                  © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
II.      Modern CFD Codes                                                                  (theoretical basis)


                           Formulation of field equations


             Eulerian                                                            Lagrangian
                                                                          Methods are not sufficiently
       FVM                    FEM                                         mature for engineering
                                                                          applications.
 Commercial codes:   Commercial codes:
 -Fluent                                                                  -Lattice Boltzmann Method
                     -COMSOL
 -CFX                                                                     -DSMC
 -CFD++              -ADINA
 -FLACS                                                                   -SPH
 -FloTHERM
 -COSMOS             Open-Source:
 -EFDLab
                     -Elmer                                               No commercial codes.
 -FlowVision
 -PHOENIX                                                                 Only “experimental”
 -… many more !                                                           Open-Source codes !
 Open-Source:
 -OpenFOAM
 -Code_Saturne
 -FDS
 -OpenFVM




                     © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
II. Modern CFD Codes                                                                 (building blocks)

                                       What to :
  Building Blocks

   Geometry &
                                       Presume                  • Boundary / initial conditions
                                                                • Physical models

  grid generation
                                       and what to :
       module                                                   • Physical phenomena (i.e. heat

                                       Neglect                    transfer, chemical reaction…etc)
                                                                • Properties (i.e. compressibility, real
  Problem settings                                                gas…etc)

      module
                                       and what to :


                                       Calculate
                                                                • Unknowns
                                                                • Relationships between field variables
  Solution module


    Visualization                         Solution methods and algorithms
       module                             • Spatial / temporal descritization schemes
                                          • Convergence criterion
                                          • Interpolation methods
                                          • …etc.

               © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
II.       Modern CFD Codes                                               (Open-source vs. commercial)

      Feature                        Open-source                         Commercial
      Customization of                      Excellent                            Good
      physical models

      Customization of                      Excellent                            Poor
      numerical schemes

      Authenticity of                       Excellent                            Good
      physical models
      User friendliness                        Poor                           Excellent

      Free technical                           Poor                              Good
      support
      Paid technical                        Excellent                         Excellent
      support
      Academic use                        Excellent                      Poor ~ Good

      Industrial use                          Poor                           Excellent




                       © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
III. Modern Applications of CFD
• Industrial sector (examples)
  – Electronics cooling




  – Chemical engineering




          © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
III. Modern Applications of CFD
• Medicine and biomechanics

Vascular homodynamic

Respiratory system
aerodynamics

Air-breathing disease
transmission




Velocity magnitude plotted on a midplane slice of the abdominal aorta for one representative
subject under resting (left) and simulated exercise (right) conditions.



                      © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
III. Modern Applications of CFD
• Medicine and biomechanics

Vascular homodynamic

Respiratory system
aerodynamics


Airborne transmission of
respiratory infectious diseases




Deposition enhancement factors (DEFs) for a steady state inhalation flow rate of 30 L/min in a
cast-based model of the upper TB airways with particle sizes of (a) 40 nm and
(b) 4 μm. DEF values represent the ratio of locally deposited mass to total deposited mass in
the airway. Significant enhancement in local deposition is observed for both nanometer
(factor of 25) and micrometer (factor of 110) aerosols.

                       © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
III. Modern Applications of CFD
• Medicine and biomechanics

Vascular homodynamic

Respiratory system
aerodynamics


Airborne transmission of
respiratory infectious diseases




Airflow streamlines of supplied and exhaled airflows in Scenario #1: (a) supplied airflow; (b)
exhaled airflow of P8; (c) exhaled airflow of P12; (d) exhaled airflow of P21; (e) exhaled
airflow of P27; (f) exhaled airflow of P30.


                       © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
III. Modern Applications of CFD
• Homeland security

   – Simulation of
     atmospheric dispersion
     of a contaminant
     released in downtown
     Portland, Oregon. The
     wind is from the west
     and the contaminant is
     released at the small
     yellow “+.”

     Annu. Rev. Fluid Mech.
     2006. 38:87–110




              © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
IV.    CFD for combustion applications
                                                                                      1- Deflagration
                                                            Stationary Flames
                                                            Propagating flames        2- Detonation
Classification according                Flames
to temporal development                                     Propagating Flames
                                                            Stationary flames
                                                                                      1- Laminar Flames
                                                                                      2- Turbulent Flames



                                                                                      1- Laminar Flames
                                                             Premixed Flames          2- Turbulent Flames
                                                                                      3- Propagating Flames
Classification according                                                                              1- Laminar Flames
to pre-mixing between                   Flames             Non-premixed (diffusion) Flames            2- Turbulent Flames

fuel and air                                                                                          1- Laminar Flames
                                                               Partially-premixed Flames              2- Turbulent Flames




                                                                                              Bluff-body stabilization
                                                                Stable Flames
                                                                                             Aerodynamic stabilization
Classification according                                                                    1- Swirl stabilized flames
                                                                                            2- Vortex stabilized flames
to stability criterion                  Flames                                              3- Flow field stabilized flames

                                                                                        1- Thermal instability
                                                                                        2- Aerodynamic instability
                                                               Unstable Flames          3- Acoustic instability
                                                                                        4- Chemical instability
                                                                                        5- Electromagnetic instability




                © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
IV.       CFD for combustion applications
•   It is a fact that most of engineering combustion systems involve turbulent
    flames
•   It is also a fact that any turbulent flow problem represent a challenge for CFD
    !
•   The bottleneck of CFD combustion solvers is the modeling of turbulence-
    chemistry interaction

                                                                                      Chemical
         Modeling                                                                      kinetics
         problems                                                                     problems

      Turbulence (?)                       Mixing                         Chemical reaction (?)

                           Time scale problem ?
                                   Damköhler number

               Density
               Pressure
                                                                            Heat
               Thermal conductivity
               Viscosity



                    © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
IV.       CFD for combustion applications

                          Turbulence modeling approaches



      Isotropic turbulence                        DNS                        Anisotropic turbulence

  • Algebraic models                                                         • Reynolds stress models
  • Eddy viscosity models (2                                                 • Filtering approaches (LES,
  equations, 4 equations)                                                    DES, VLES)
                                            Coupling
                                          Technique                                       • PDF model
  • EDM
                                         • Flamelet model                                 • Detailed chemical
  • EDC                                                                                   kinetics
 Eddy break-up theory                      Flamelet theory                            Detailed chemistry


                       Chemical Reaction modeling approaches



                    © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
IV. CFD for combustion applications
• Modeling of vortex flames


      Vortex flame




      Swirling flame
  1. Enhanced flame stability                     Applications:
                                                  - Micro gas turbines
    2. Shorter flame length
                                                  - Micro jet engines for miniaturized
                                                  UAVs
 3. Added flame controllability



               © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
IV.     CFD for combustion applications

Problems that had to be                                                                      Secondary
                                                                                             recirculation
   solved:                                                                                   zones
1. Turbulence modeling of
   the vortex flow field in
   this complex geometry
2. Find suitable parameters
   to study the aerodynamic
                                                                              vortex field
   stability mechanism of
   the flame
3. Conduct a parametric
   study to reveal the major
   characteristics of the
   flame




                © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
IV.     CFD for combustion applications
•   Major contributions:
    1. Formulation of the Rε/k-ε turbulence model
    2. The new model variant has been coded into Fluent and
       OpenFOAM, tested and validated for several cases of rotation
       flows
    3. Complete description of all main-flow phenomena in the
       asymmetric vortex combustor
    4. Identification of the aerodynamic stability mechanism of vortex
       flames


      Stoichiometric
      Reaction Zones




                       © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
V.      Contemporary Challenges
• Physical models

     Turbulence                          Multiphase flow                                  Chemical reaction

- Highly strained                   - Interaction                                  - Optimize between
flows                               between primary and                            physical assumptions
                                    secondary phase                                and correct chemical
- Atmospheric flow
                                                                                   kinetics
modeling                            - Integration with
                                    new turbulence                                 - Efficient coupling
- Transitional flow
                                    models                                         with turbulence
regimes
                                                                                   models




                    © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
V.                                Contemporary Challenges
• Computational requirements
     – Faster processors and larger memories
     – Smarter software algorithms
     Computational requirements                                                                             DNS




                                                                      DES
                                                       LES



                                   RANS



                                             Turbulence models (example)

                                      © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
V.     Contemporary Challenges
• Benchmarking and validation

                            Error sources in CFD



     Physical models                                                 Numerical schemes


 Encountered by                                                  Encountered by grid
 benchmarking and                                                independency studies
 validation
 Challenges:
 1- Unavailability of sufficient experimental data
 for some problems (i.e. biomechanics)
 2- Generalization of the validation procedure


                © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
VI. Future trends in CFD research
• Mesh-free methods                                 (late 1970s – today)
  – Smoothed Particle Hydrodynamics (SPH)
  – Diffuse Element Method (DEM)


• Hybrid Methods
  – Particle in Cell (PIC)
  – Lattice Boltzmann Method (LBM)
  – Direct Simulation Monte Carlo (DSMC)




          © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
VI. Future trends in CFD research
• Low-cost HPC                           Clustering concept
      Old / Unused PCs


                                                                          Linux OS


           Network

                                                                  Parallel CFD code
    HPC




            © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
VI. Future trends in CFD research
• Runtime validation concept (LES, DES)

   Problem setup                                               Problem setup            Fine tuning


     Solution                                                                             Basic
                                                                  Pre-solution          Validation
                               Change
      Results                  Model(s)
                                                                     Solution


     Validation            invalid                                    Results
   valid
                                                                  Final results
    Final results




                  © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
Thank you




© All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.

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CFD : Modern Applications, Challenges and Future Trends

  • 1. Computational Fluid Dynamics Modern Applications, Challenges and Future Trends Arab Academy for Science, Technology and Maritime Transport, 28th March 2012 Dr. Khalid M. Saqr B.Sc (Alex. Uni.), M.Eng (UTM), PhD (UTM) © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved. 28/03/2012
  • 2. Outlines I. CFD in a nutshell • Tool or an engineering science? • Frame of reference of CFD II. Modern CFD codes • Building blocks of modern codes • Commercial vs. open source codes III. Modern applications of CFD • Industrial sector • Medicine and biomechanics • Homeland security IV. CFD for combustion applications • Modeling of turbulent vortex flames V. Contemporary challenges • Physical models • Computational requirements • Benchmarking and validation VI. Future trends • Mesh-free methods • Low-cost HPC • Runtime validation concept © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 3. I. CFD in a nutshell (definition) ? CFD: An engineering tool…? Or an independent branch of engineering science ? Computational homodynamic Computational Oceanography 2nd World War weaponry 1930s – 1940s 2000 – 2010s © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 4. I. CFD in a nutshell (definition) Computer Science Fluid Mechanics & Software Engineering Heat Transfer CFD Electromagnetic field Processor development Chemical reaction Parallel computing Physical & Life Sciences Engineering science Geophysics Vascular medicine Oceanography 21st Century definition of CFD: Solving the governing equations of fluid flow -by using computational methods- under various physical conditions (e.g. heat transfer, radiation, electromagnetic field, nuclear reaction…etc.) to study different phenomena encountered in the numerous branches of physical science. © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 5. I. CFD in a nutshell (frame of reference) Frame of reference of engineering CFD Theories of Continuum Molecular / Quantum theory matter: hypothesis Kinetic theory Mechanics: Newtonian Relativistic Chaos theory: Deterministic Random chaos chaos 1. Matter (i.e. fluid) is assumed to be a continuum, and its properties and characteristics can be described by field functions. 2. These field functions are functions in Euclidian space and time, and their frame of reference is an inertial one. 3. The sensitive dependence on initial conditions described by the governing partial differential equation does not involve any random events © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 6. II. Modern CFD Codes (theoretical basis) Formulation of field equations Eulerian Lagrangian Methods are not sufficiently FVM FEM mature for engineering applications. Commercial codes: Commercial codes: -Fluent -Lattice Boltzmann Method -COMSOL -CFX -DSMC -CFD++ -ADINA -FLACS -SPH -FloTHERM -COSMOS Open-Source: -EFDLab -Elmer No commercial codes. -FlowVision -PHOENIX Only “experimental” -… many more ! Open-Source codes ! Open-Source: -OpenFOAM -Code_Saturne -FDS -OpenFVM © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 7. II. Modern CFD Codes (building blocks) What to : Building Blocks Geometry & Presume • Boundary / initial conditions • Physical models grid generation and what to : module • Physical phenomena (i.e. heat Neglect transfer, chemical reaction…etc) • Properties (i.e. compressibility, real Problem settings gas…etc) module and what to : Calculate • Unknowns • Relationships between field variables Solution module Visualization Solution methods and algorithms module • Spatial / temporal descritization schemes • Convergence criterion • Interpolation methods • …etc. © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 8. II. Modern CFD Codes (Open-source vs. commercial) Feature Open-source Commercial Customization of Excellent Good physical models Customization of Excellent Poor numerical schemes Authenticity of Excellent Good physical models User friendliness Poor Excellent Free technical Poor Good support Paid technical Excellent Excellent support Academic use Excellent Poor ~ Good Industrial use Poor Excellent © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 9. III. Modern Applications of CFD • Industrial sector (examples) – Electronics cooling – Chemical engineering © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 10. III. Modern Applications of CFD • Medicine and biomechanics Vascular homodynamic Respiratory system aerodynamics Air-breathing disease transmission Velocity magnitude plotted on a midplane slice of the abdominal aorta for one representative subject under resting (left) and simulated exercise (right) conditions. © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 11. III. Modern Applications of CFD • Medicine and biomechanics Vascular homodynamic Respiratory system aerodynamics Airborne transmission of respiratory infectious diseases Deposition enhancement factors (DEFs) for a steady state inhalation flow rate of 30 L/min in a cast-based model of the upper TB airways with particle sizes of (a) 40 nm and (b) 4 μm. DEF values represent the ratio of locally deposited mass to total deposited mass in the airway. Significant enhancement in local deposition is observed for both nanometer (factor of 25) and micrometer (factor of 110) aerosols. © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 12. III. Modern Applications of CFD • Medicine and biomechanics Vascular homodynamic Respiratory system aerodynamics Airborne transmission of respiratory infectious diseases Airflow streamlines of supplied and exhaled airflows in Scenario #1: (a) supplied airflow; (b) exhaled airflow of P8; (c) exhaled airflow of P12; (d) exhaled airflow of P21; (e) exhaled airflow of P27; (f) exhaled airflow of P30. © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 13. III. Modern Applications of CFD • Homeland security – Simulation of atmospheric dispersion of a contaminant released in downtown Portland, Oregon. The wind is from the west and the contaminant is released at the small yellow “+.” Annu. Rev. Fluid Mech. 2006. 38:87–110 © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 14. IV. CFD for combustion applications 1- Deflagration Stationary Flames Propagating flames 2- Detonation Classification according Flames to temporal development Propagating Flames Stationary flames 1- Laminar Flames 2- Turbulent Flames 1- Laminar Flames Premixed Flames 2- Turbulent Flames 3- Propagating Flames Classification according 1- Laminar Flames to pre-mixing between Flames Non-premixed (diffusion) Flames 2- Turbulent Flames fuel and air 1- Laminar Flames Partially-premixed Flames 2- Turbulent Flames Bluff-body stabilization Stable Flames Aerodynamic stabilization Classification according 1- Swirl stabilized flames 2- Vortex stabilized flames to stability criterion Flames 3- Flow field stabilized flames 1- Thermal instability 2- Aerodynamic instability Unstable Flames 3- Acoustic instability 4- Chemical instability 5- Electromagnetic instability © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 15. IV. CFD for combustion applications • It is a fact that most of engineering combustion systems involve turbulent flames • It is also a fact that any turbulent flow problem represent a challenge for CFD ! • The bottleneck of CFD combustion solvers is the modeling of turbulence- chemistry interaction Chemical Modeling kinetics problems problems Turbulence (?) Mixing Chemical reaction (?) Time scale problem ? Damköhler number Density Pressure Heat Thermal conductivity Viscosity © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 16. IV. CFD for combustion applications Turbulence modeling approaches Isotropic turbulence DNS Anisotropic turbulence • Algebraic models • Reynolds stress models • Eddy viscosity models (2 • Filtering approaches (LES, equations, 4 equations) DES, VLES) Coupling Technique • PDF model • EDM • Flamelet model • Detailed chemical • EDC kinetics Eddy break-up theory Flamelet theory Detailed chemistry Chemical Reaction modeling approaches © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 17. IV. CFD for combustion applications • Modeling of vortex flames Vortex flame Swirling flame 1. Enhanced flame stability Applications: - Micro gas turbines 2. Shorter flame length - Micro jet engines for miniaturized UAVs 3. Added flame controllability © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 18. IV. CFD for combustion applications Problems that had to be Secondary recirculation solved: zones 1. Turbulence modeling of the vortex flow field in this complex geometry 2. Find suitable parameters to study the aerodynamic vortex field stability mechanism of the flame 3. Conduct a parametric study to reveal the major characteristics of the flame © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 19. IV. CFD for combustion applications • Major contributions: 1. Formulation of the Rε/k-ε turbulence model 2. The new model variant has been coded into Fluent and OpenFOAM, tested and validated for several cases of rotation flows 3. Complete description of all main-flow phenomena in the asymmetric vortex combustor 4. Identification of the aerodynamic stability mechanism of vortex flames Stoichiometric Reaction Zones © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 20. V. Contemporary Challenges • Physical models Turbulence Multiphase flow Chemical reaction - Highly strained - Interaction - Optimize between flows between primary and physical assumptions secondary phase and correct chemical - Atmospheric flow kinetics modeling - Integration with new turbulence - Efficient coupling - Transitional flow models with turbulence regimes models © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 21. V. Contemporary Challenges • Computational requirements – Faster processors and larger memories – Smarter software algorithms Computational requirements DNS DES LES RANS Turbulence models (example) © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 22. V. Contemporary Challenges • Benchmarking and validation Error sources in CFD Physical models Numerical schemes Encountered by Encountered by grid benchmarking and independency studies validation Challenges: 1- Unavailability of sufficient experimental data for some problems (i.e. biomechanics) 2- Generalization of the validation procedure © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 23. VI. Future trends in CFD research • Mesh-free methods (late 1970s – today) – Smoothed Particle Hydrodynamics (SPH) – Diffuse Element Method (DEM) • Hybrid Methods – Particle in Cell (PIC) – Lattice Boltzmann Method (LBM) – Direct Simulation Monte Carlo (DSMC) © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 24. VI. Future trends in CFD research • Low-cost HPC Clustering concept Old / Unused PCs Linux OS Network Parallel CFD code HPC © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 25. VI. Future trends in CFD research • Runtime validation concept (LES, DES) Problem setup Problem setup Fine tuning Solution Basic Pre-solution Validation Change Results Model(s) Solution Validation invalid Results valid Final results Final results © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.
  • 26. Thank you © All copyrights belong to Dr. Khalid M. Saqr. All rights reserved.