A physical approach for development of computational algorithms A physical approach for development of computational algorithms for solving the Navierfor solving the Navier--Stokes equations Stokes equations and its application in jet engine analysisand its application in jet engine analysis

Contents:

1) Main difficulties of the efficient algorithm development

2) Simulation of incompressible flowsa) description of unified algorithmb) solution of benchmark problemsc) solution of applied problems

3) Simulation of compressible flowsa) strong nonlinear nature of compressible Navier-Stokes equationsb) flow in Laval nozzle

4) Flows with determinate mass flow rate

Marу L. Yanovskaya, Sergey I. MartynenkoFSUE Central Institute of Aviation Motors n.a. P.I. Baranov, Moscow, Russia

TwoTwo--dimensional incompressible Navierdimensional incompressible Navier--Stokes equationsStokes equations

Main difficulties of numerical solution of the NavierMain difficulties of numerical solution of the Navier--Stokes equationsStokes equations

1) Nonlinear nature of the Navier-Stokes equations2) Singular perturbation of the momentum equations3) Structure of the Navier-Stokes equations

The system cannot be solved by standard methods because of a33=0.

Main difficulties of the pressure computation1) Absence of equation for the pressure computation2) Absence of boundary conditions for the pressure correction3) Required high accuracy of pressure computation

Segregated (decoupled) algorithms Segregated (decoupled) algorithms

Method SIMPLE (Method SIMPLE (SpaldingSpalding B.B., Patankar S., 1972), Patankar S., 1972)

Main disadvantage of similar methods consists of the artificial boundary conditions for pressure (correction).

Method of artificial Method of artificial compressibilitycompressibility ((ChorinChorin AA., 1967)., 1967)

The method does not require some artificial boundary conditions for pressure. However convergence rate is slow.

Coupled algorithmsCoupled algorithms

It is very difficult to apply the coupled methods for computations of compressible flows.

Simplified NavierSimplified Navier--Stokes equationsStokes equations

Since p=p(x), very efficient numerical methods for simplified Navier-Stokes equations have been proposed and developed (for example, the secant method of Briley W, 1974).

Basic idea: the pressure decompositionBasic idea: the pressure decomposition

Steady flow in cavity (no directed fluid flows)Steady flow in cavity (no directed fluid flows)

Predictor (Predictor (auxiliaryauxiliary problem)problem)

Corrector (main problem)Corrector (main problem)

Structure of algorithmStructure of algorithm(mass flow rate is given)(mass flow rate is given)

1. Auxiliary problem2. Main problem3. Check convergence: return to item 1 (if necessary)

Steady flow in cavitySteady flow in cavity ((Re=100Re=100, , staggered grid staggered grid 101101××101)101)

Classical decoupled algorithm:

Proposed algorithm:Proposed algorithm:

Distribution of Distribution of uu velocity component in vertical sectionvelocity component in vertical section

(0.5, )

0

1

0.5

Y

0 0.5

1

Yu

xvX

1

( , 0.5)

X

1

y

Reduction of residual of the continuity equationReduction of residual of the continuity equation

-0.115

-0.1

-0.1

-0.1-0.065

-0.065

-0.065

-0.065

-0.065

-0.015

-0.015

-0.015

-0.015

-0.015

-0.015

-0.015

-0.001

0.001

-0.001

-0.001 -0.001

-0.0

-0.001

-0.001

5E-05

5E-05

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Steady flow in cavitySteady flow in cavity ((Re=100Re=1000, 0, staggered grid staggered grid 301301××3301)01)Isolines of the stream function

Steady flow in cavitySteady flow in cavity ((Re=100Re=1000, 0, staggered grid staggered grid 301301××3301)01)Isobars (p(0,0)=0)

-0.111

-0.105

-0.105

-0.09

-0.09-0.09

-0.09

-0.07

-0.07-0.07

-0.07

0.05

-0.05

-0.05

-0.05

-0.05

-0.05

-0.03

-0.03

-0.03

-0.03-0.03

-0.03

-0.01-0.01

-0.01

-0.005

-0.005

-0.005

0

0

0

0.2

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Velocity components in middle sections (Re=1000)Velocity components in middle sections (Re=1000)

(0.5, )u y0.6

0.20.0-0.4

-0.2

0.2

0.0

0.4

y0.4 0.6 0.8

0.8

1.0

( , 0.5)-0.4

-0.61.0 0.0 0.2

0.0

-0.2 v

0.60.4 x 0.8

x

1.0

0.4

0.2

Reduction of the computational efforts is ∼ 30…50%.

Remarks on the predictor Remarks on the predictor

1) Equations of auxiliary problem are solved by numerical methods developed for simplified Navier-Stokes equations.

2) Mass conservation equations are considered as a priori information about velocity components.

3) Velocity and “pressure” components (px, py, pz ) are computed in coupled manner.

4) Local nature of the algorithm for solving the auxiliary problem (without global linearization).

Steady backward-facing step flow (Re=800, staggered grid 101 ×1401)

Reduction of the computational efforts is in ∼ 400 times.

Steady backward-facing step flow (Re=800, staggered grid 101 ×1401)Isolines of the stream function

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

Steady backward-facing step flow (Re=800, staggered grid 101 ×1401)Isobars

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

Applied problemApplied problem: : flow in catalyst of micro liquidflow in catalyst of micro liquid--propellant enginepropellant engine

Catalyst of micro liquidCatalyst of micro liquid--propellant enginepropellant engine

1 2 31

catalyst needle(iridium)

axis of symmetry

solid wall

29.2 10 m-6

.

15 10 m.

-6

300 10 m-6. 3000 10 m. -6

250

10

m.

-6

Staggered grid in catalyst of micro liquidStaggered grid in catalyst of micro liquid--propellant engine propellant engine (385(385××3150)3150)

IsolinesIsolines of the of the streamstream functionfunction ((Re=350)Re=350)

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.850.00

0.10

0.20

0.30

IsolinesIsolines of the of the streamstream functionfunction ((Re=350)Re=350)

4.40 4.50 4.600.00

0.10

0.20

0.30

Isobars (Re=350)Isobars (Re=350)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.850.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Weak compressible flowsWeak compressible flowsH=500·10-6m

Compressible flowsCompressible flows

FLUENT softwareFLUENT software

FLUENT softwareFLUENT software

FLUENT softwareFLUENT software

Compressible flow in driven cavityCompressible flow in driven cavity

Compressible flow in driven cavity: predictorCompressible flow in driven cavity: predictor

Compressible flow in driven cavity: correctorCompressible flow in driven cavity: corrector

Simulation of compressible flow in Laval nozzleSimulation of compressible flow in Laval nozzle

BodyBody--fitted coordinates fitted coordinates

Mapping of physical domain onto computational domainMapping of physical domain onto computational domain

Draft grid in Laval nozzleDraft grid in Laval nozzle

Draft grid in Laval nozzle near throatDraft grid in Laval nozzle near throat

XX--direction velocity [m/s]direction velocity [m/s]

X

Y

0 0.0005 0.001 0.0015 0.002 0.0025 0.0030

0.0005

0.001

X Velocity: 100 400 700 1000

XX--direction velocity [m/s] near direction velocity [m/s] near throatthroat

X

Y

0.0004 0.0005 0.0006 0.0007 0.000

0.0001

0.0002

0.0003

X Velocity: 100 400 700 1000

Pressure drop PPressure drop P00--P [Pa]P [Pa]

X

Y

0 0.0005 0.001 0.0015 0.002 0.0025 0.0030

0.0005

0.001

Pressure: -95000 -65000 -35000 -5000

Pressure drop PPressure drop P00--P [Pa] near P [Pa] near throatthroat

X

Y

0.0004 0.0005 0.0006 0.0007 0.000

0.0001

0.0002

0.0003

Pressure: -95000 -65000 -35000 -5000

Static temperature [K]Static temperature [K]

X

Y

0 0.0005 0.001 0.0015 0.002 0.0025 0.0030

0.0005

0.001

Temperature: 50 100 150 175 200 250 300 350 400 450

Static temperature [K] near Static temperature [K] near throatthroat

X

Y

0.0004 0.0005 0.0006 0.0007 0.000

0.0001

0.0002

0.0003

Temperature: 50 100 200 250 275 300 350 375 385 400 425 450

Distribution pressure [Pa] along the nozzle axisDistribution pressure [Pa] along the nozzle axisthroat

0 0.001 0.002 0.003 0.004x, m

0x100

2x104

4x104

6x104

8x104

1x105

Pres

sure

, Pa

outlet

Flows with determinate mass flow rate Flows with determinate mass flow rate

Geometry of problem about moving plunger

1. Main problem 2. Check convergence: continue if necessary3. Auxiliary problem

Structure of algorithmStructure of algorithm(mass flow rate is determined)(mass flow rate is determined)

Flows with Flows with determinateddeterminated mass flow rate mass flow rate

Isolines Isobars

Re=200Re=200, , staggered grid 2staggered grid 20101××24240101

CONCLUSIONSCONCLUSIONS

Unified algorithm for solving (in)compressible Navier-Stokes equationsin primitive variables is proposed and developed. The algorithm uses physical aspects of hydrodynamics for reduction of computationalefforts. As a result, the algorithm can be incorporated with anymathematical method for solving Navier-Stokes equations. Impressive reduction of computational work is observed for directed fluid flows.

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