Transcript Praca magisterska pt.: „Systemy eksploracji danych w

Modelling of laminar flow using
Numerical Methods
Marta Korolczuk-Hejnak
AGH University of Science and Technology in Krakow,
Faculty of Metal Engineering and Industrial Computer Science,
Department of Ferrous Metallurgy
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Kraków, 08.12.2010 r.
Content
•
Primary definitions
•
Types of flow
•
Reynolds number
•
Navier – Stokes equations
•
Numerical solutions methods used in flow problems
•
Navier – Stokes solution by FDM for laminar flow
•
Numerical results get by FDM and FEM methods for laminar
flow
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Fluid, flow - definitions
Fluid
A continuous, amorphous substance (liquid or gas) whose molecules move freely past
one another and that has the tendency to assume the shape of its container.
Flow
The motion of the fluid
Types of flow:

Laminar flow

Transitional flow

Turbulent flow
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Laminar flow
 occurs when a fluid flows in parallel layers, with no disruption between the layers,
 steady-state -
,
(1)
 in nonscientific terms laminar flow is "smooth," „orderly”
 generally happens when dealing with small pipes and low flow velocities;
can be regarded as a series of liquid cylinders in the pipe, where the innermost parts
flow the fastest, and the cylinder touching the pipe isn't moving at all,
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Pic.1. Laminar flow
Pic.2. Velocity distribution in the pipe for
laminar flow
Turbulent flow
 characterized by chaotic, stochastic property changes,
 unsteady – state flow -
(2)
 in nonscientific terms turbulent flow is „rough, „random” , „chaotic”
 vortices, eddies and wakes make the flow unpredictable;
happens in general at high flow rates and with larger pipes,
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Pic.3. Turbulent flow
Pic.4. Velocity distribution in the pipe for
turbulent flow
Transitional flow
 situation as the flow speed was increased the dye fluctuates and one observes
intermittent bursts
 mixture of laminar and turbulent flow, with turbulence in the center of the pipe, and
laminar flow near the edges;
each of these flows behave in different manners in terms of their frictional energy loss
while flowing, and have different equations that predict their behavior.
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Pic.5. Transitional flow
Reynolds number
Reynolds number Re
Dimensionless number gives a measure of the ratio of inertial forces ρV2/L to viscous
forces μV/L2 and consequently quantifies the relative importance of these two types of
forces for given flow conditions
(3)
• V – mean fluid velocity, m/s
• L – characteristic linear dimension (traveled lenght of fluid), m
• μ – dynamic viscosity of the fluid, Pa·s
• τ – shear stress, Pa
• - shear rate, 1/s
• υ- kinematic viscosity of the fluid, m^2/s
• ρ – density of the fluid, kg/m^3
For flow in a pipe of diameter D, experimental observations show that:
• laminar flow Re < 2300,
• transitional flow 2300<Re < 4000,
• turbulent flow Re >4000.
(4)
(5)
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Navier-Stokes equations
 named after Claude-Louis Navier * and and George Gabriel Stokes**, describe the
motion of fluid substances,
[* Claude Louis Marie Henri Navier (10 February 1785 in Dijon – 21 August 1836
in Paris) born was a French engineer and physicist who specialized in
mechanics ],
[** Sir George Gabriel Stokes (13 August 1819 Skreen, County Sligo, Ireland – 1 February 1903 Cambridge, England), was a mathematician and physicist
who made important contributions to fluid dynamics, optics, and mathematical
physics ],
 describe the physics of many things of academic and economic interest;
may be used to model the weather, ocean currents, water flow in a pipe, air flow
around a wing, and motion of stars inside a galaxy, design of aircraft and cars, the study
of blood flow, the design of power stations, the analysis of pollution etc.,
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Navier-Stokes equations
used for mathematical characteristic of flow phenomenons in a system with known geometry,
 arise from applying:
o Newton's second law to fluid motion,
o assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the
gradient of velocity),
o pressure term,
general form of the equations of fluid motion
(6)
(7)
(7)
• u – flow velocity vector ,
• ρ – fluid density,
• p – pressure,
• S - deviatoric, stress tensor,
• g – gravitation acceleration,
• μ – dynamic viscosity of the fluid,
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Numerical solutions
 Numerical approximation methods used for solving
differential equations:
o FDM (polish MRS) – Finite Difference Method,
• Curvilinear Finite Difference ,
o FEM (polish MES) – Finite Element Method,
o BEM - Boundary Element Method ,
o FVM (polish MOS) – Finite Volume Method
o NI (polish CN) – Numerical Integration.
Steps in FDM:
o Aproximate the solutions to differential equations by
replacing derivative expressions with aproximately
equivalent difference quatients.
 Steps in FEM:
o Finding aproximate solutions of partial differential
equations as well as of integral equations:
I.
Discretization of the domain into a set of finite
elements.
II. Defining an approximate solution over the
element.
III. Weighted integral formulation of the differential
equation.
IV. Substitute the approximate solution and get the
algebraic equation.
Pic.6. Schematic of finding the solution
using numerical methods [2]
 Steps in FVM:
o
Represanting and evaluating partial differential
equations in the form of algebraic equations.
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Numerical solutions in FDM
Non-dimensional equations of Navier-Stokes.
2nd – continouity equation must be true during the whole simulation.
(8)
(9)
Simple (primitive) variables:
u = (u,v) - velocity vector,
p - pressure
(10)
(11)
(12)
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(P*) ^n- initial value of pressure field,
(U*)^n, (V*)^n- velocity fields
Pressure correction (using
Poisson equation):
(13)
Pic.7. Schematic of finding the solution using
the SIMPLE algorithm [4]
Nabla operator, divergence
operator
(14)
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Pic.9. Schematic of grid used
in the SIMPLE algorithm [4]
• dark points – pressure p,
• white points – x - direction component of velocity
u,
• cross – y - direction component of velocity v
Pic.8. Schematic of discretization used
in the SIMPLE algorithm [4]
- Front difference quention
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- Central difference quention
Schematic of the discretization FDM
(15)
(16)
Discrete equations
(17)
(18)
(25)
(26)
(19)
(27)
(28)
(20)
(21)
(22)
(23)
(24)
Pressure Poisson equation FDM
(29)
(30)
(31)
Pic.10. Schematic of grid used
for pressure solutions in
the SIMPLE algorithm [4]
(32)
(33)
If value of the difference between ‘old’ and ‘new’
value of pressure field is < than ε -> FINISH the
procedure.
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Numerical solutions by FDM
Pic.10. Streamlilnes in a lid-driven
cavity for Re = 400 [4]
Pic.11. Fluid flow in a 3- interspace
channel for Re= 10 [4]
•Red color – field of plane velocity
•Green color –filed of perpendicular
velocity
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Numerical solutions by FEM
FORMULATION FOR ISOTHERMAL, LAMINAR FLOW
•Example 1 : Fully developed laminar flow in a two dimensional rectangular
channel.
Pic.12. Boundary conditions
Fully developed flow in a rectangular
channel [3]
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Numerical solutions by FEM
FORMULATION FOR ISOTHERMAL, LAMINAR FLOW
Pic.13. Pressure contours for Re=1
Fully developed flow in a rectangular channel [3]
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Numerical solutions by FEM
FORMULATION FOR ISOTHERMAL, LAMINAR FLOW
•Example 2 : Flow in a lid-driven cavity.
Pic.14. Boundary conditions and finite element mesh (41×41) for flow
in a lid-driven cavity [3]
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•Example 2 : Flow in a lid-driven cavity.
Re=1
Pic.15-16. Streamlilnes and pressure contours at steady state for flow
in a lid-driven cavity [3]
Re=100
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•Example 3 : Flow in a backward step.
Re=400
Pic.17-18. Streamlilnes and pressure contours at steady state for flow
in a lid-driven cavity [3]
Re=1000
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References:
1. J.G.Heywood, K. Masuda, R. Rautmann, V.A. Solonnikov, „The Navier-Stokes
Equations Theory and Numerical Methods”, Springer-Verlag, 1988, Oberwolfach .
2. M. Kmiotek, „ Przegląd solverów numerycznych stosowanych w mechanice
obliczeniowej”, Scientific Bulletin of Chelm, Section of Mathematics and Computer
Science, No. 1/2008.
3. R.W .Lewis. , K. Ravindran and A.S. Usmani, „Finite Element Solution of
Incompressible Flows Using an Explicit Segregated Approach”, Archives of
Computational Methods in Engineering, Vol. 2, 4, 69–93 (1995).
4. M. Matyka, „Hydro-dynamica Rozwiązania numerycne równań przepływu cieczy
nieściśliwych”, http://panoramix.ift.uni.wroc.pl/~maq
5. A.T. Patera, „ A spectral element method for fluid dynamics: Laminar flow in a
channel expansion”, Journal of Computationing Physics 54, 468-488 (1984).
6. R.Peyret, T.D. Taylor, „Computational Methods for Fluid Flow”, Springer-Verlag New
York Inc., 1983, USA.
7. O.C. Zienkiewicz, R.L. Taylor, „The finite element method Volumev 3 Fluid
Dynamics”, Fifth Edition, Butterworth-Heinemann ,Oxford, 2000
8. O.C. Zienkiewicz, „The finite element method” Fourth Edition Volume 1 Basic
Formulation and Linear Problems, McGraw-Hill International (UK), 1989, Londyn.
9. O.C. Zienkiewicz, „The finite element method” Fourth Edition Volume 2 Solid and
Fluid Mechanics Dynamics and Non-linearity, McGraw-Hill International (UK), 1991,
Londyn.
10. www. wikipedia.org
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Thank you for your attention.
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