Hydrostatics Balance equation Mass balance Momentum balance Bernoulli s equation Energy balance Classification of PDE Examples

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1 Hdrostatics Balance eqation Mass balance Momentm balance Bernolli s eqation Energ balance Classification of DE Eamples καθ. Γ.Μπεγελές Balance Eqations /.1

2 Hdrostatic eqation d d,, ) dd,, ) dd gddd d d,, ),, ) g d d g.1a) d grad g a καθ. Γ.Μπεγελές 0 Balance Eqations /.

3 Forces on srfaces and essels - 0 gh.3) καθ. Γ.Μπεγελές Balance Eqations /.3

4 ascal s la/hdralics F A B B B A A B F F F F A B B A > A F A B A δ καθ. Γ.Μπεγελές S S A B F B δ B Balance Eqations /.4

5 0 1 0 Hdrostatic eqation for compressible flids γ γ 1 g C 0 g C h h.6).5) καθ. Γ.Μπεγελές Balance Eqations /.5

6 Kinematics of Flids,, ) dr dt d d,, dt dt a d dt dφ dt Φ t d dt grad) Φ a a a t t t t t t t t a grad) t καθ. Γ.Μπεγελές d dt t grad t Balance Eqations /.6

7 Balance Eqations /.7 Rate of deformation e e, e ) ) ) k j i k j i rot rot 1 ω καθ. Γ.Μπεγελές

8 Balance Eqations /.8 Neton s la for shear stress ij µ µ µ µ µ ) ) ) Netonian and non-netonian flids καθ. Γ.Μπεγελές

9 Balance Eqations {Rate of Φ accmlated ithin the control olme } {rate of Φ or Φ fl in nit time) entering the control olme throgh the eternal srface} -{rate of Φ eiting the control olme throgh the eternal srface} Sorces or Sinks per nit time καθ. Γ.Μπεγελές Balance Eqations /.9

10 Mass balance,, ) ) ψ, ) 3 3 Mass entering the olme in time dt throgh srfaces AD and DC) d /,, t) ddt, d /, t) ddt Mass leaing the control olme in time dt throgh srfaces BC and AB) d /,, t) ddt, d /, t) ddt Mass accmlating ithin the olme dd) dd) t dt t t ) ) καθ. Γ.Μπεγελές 0 0 Balance Eqations /.10

11 Conseration of linear momentm {Rate of momentm accmlated ithin the control olme} {rate of momentm or momentm fl) entering the control olme throgh the eternal srface} -{rate of momentm eiting the control olme throgh the eternal srface} καθ. Γ.Μπεγελές {eternal forces acting on the mass of the control olme}. Balance Eqations /.11

12 Momentm balance he component of -direction momentm fl entering the control olme throgh srfaces AD and DC) is he corresponding component of momentm fl eiting the control olme throgh srfaces BC and AB) Graitational force Where g is the component of the graitational acceleration. he resltant pressre force in the direction de to pressre acting on srfaces AD and BC ) he resltant -direction component of force de to normal and shear stress forces in the direction on CD, AB, AD and BC srfaces) he rate of accmlation -direction momentm { d} d / { d} d / { d } d / { d} d / ddg pd pd { } d / { } d / d d d d { } d / { } d / { } d / { } d / καθ. Γ.Μπεγελές dd) t Balance Eqations /.1

13 Momentm balance in direction { d} d / { d} d / καθ. Γ.Μπεγελές Balance Eqations /.13

14 Balance Eqations /.14 Naier-Stokes eqations g p t ) ) ) g p t ) ) ) g p t g p t καθ. Γ.Μπεγελές

15 Balance Eqations /.15 arios forms of the N-S eqations g p t g p t g grad t ) rot ) ) g p rot t 1 ) 0 } { rot g καθ. Γ.Μπεγελές

16 Balance Eqations /.16 Bernolli s eqation H g 0 ) } 1 { 0 ) } 1 { d s rot d s g d s t d s rot d s g d s t h t g g C C Ε γ γ γ γ c c καθ. Γ.Μπεγελές

17 Conseration of Energ First La of hermodnamics Heat Q n DE Q Dt Work W Energ E ds d καθ. Γ.Μπεγελές Both Q and W are path fnctions process dependent) bt their difference Q-W is a point fnction de is a total differential a thermodnamic propert) W Balance Eqations /.17

18 Balance Eqations /.18 ) ) Φ µ k q Dt D e k k k q Dt D e 3 Φ Φ συνάηση αποόφησης) Energ balance καθ. Γ.Μπεγελές

19 Balance Eqations /.19 Εξίσωση μεαφοάς ενοπίας 1 d p S d e d Dt D Dt D e Dt s D ) Φ k Dt s D µ 1 S gen k Dt D s Φ K m att k S gen 3 µ καθ. Γ.Μπεγελές

20 Balance Eqations /.0 J H G F t U k e F e U ), Μηωική μοφή εξισώσεων διαήησης καθ. Γ.Μπεγελές

21 Balance Eqations /.1 k e G k e H ) q f f f f f f J 0 Μηώα G,H,J καθ. Γ.Μπεγελές

22 Κλείσιμο συσήμαος ΔΕ Οι πος επίλυση διαφοικές εξισώσεις διαήησης είναι πένε: Διαήηση μάζας 1), διαήηση ομής 3), διαήηση ενέγειας 1). Το σύσημα ων εξισώσεων διαήησης έχει πένε διαφοικές εξισώσεις με έξι αγνώσους,,,, p, ). Η εσωεική ενέγεια e ου ευσού εκφάζεαι με μία εξίσωση ης μοφής p ) e e, που για αέια απλοποιείαι σην e C με C ην ειδική θεμόηα ου αείου υπό σαθεό όγκο και Τα η θεμοκασία ου αείου. Τέλος η πίεση, η πυκνόηα και η θεμοκασία συνδέοναι με ην καασαική εξίσωση p, ) που για αέια παίνει η γνωσή μοφή R καθ. Γ.Μπεγελές Balance Eqations /.

23 A general form of the eqation t Φ) U Φ ΓΦ SΦ Characteristics mltidimensional trblent flos reqire at least 6 eqations strong copling non-linear j j Φ j Φ U H c k ε i,,,,, Γ, καθ. Γ.Μπεγελές φ S φ Balance Eqations /.3

24 -Ö- -S Ö µ µ r r r 1 µ µ µ r r r r r r r 1 Ô 0 k G-ñå å C 1 åg-c ñå )/k G r r r µ Sorce erms SΦ καθ. Γ.Μπεγελές

25 Classification of flo phenomena Stead, nstead Incompressible, compressible D, 3D General form of DE AΦ BΦ CΦ DΦ EΦ FΦG Where A,B,C,D,E,F and G fnctions of,,φ,φ,φ B -AC<,0,> arabolic, elliptic, hperbolic, partiall elliptic directional transport of information tpe of bondar conditions καθ. Γ.Μπεγελές Balance Eqations /.5

26 Model roblems Heat Condction Bondar laers Recirclating flos Spersonic flos ime dependent problems καθ. Γ.Μπεγελές Bondar ale problems Initial ale problems Balance Eqations /.6

27 Hperbolic DE b Hperbolic DEs often model ibrating sstems or ae motion. Eamples: 4ac 0 1D ae eqation t adection eqation elocit > t c 0 καθ. Γ.Μπεγελές displacement concentration Balance Eqations /.7

28 arabolic DE arbolic DEs often describe heat flo and other diffsie processes. α t transient diffsion eqation Eamples: b 4ac 0 t καθ. Γ.Μπεγελές adection diffsion eqation concentration D Balance Eqations /.8

29 Elliptic DE Elliptic DEs sall describe stead state phenomena. Laplace s eqation Eamples: b 4ac oisson s eqation < 0 f 0, ) καθ. Γ.Μπεγελές Balance Eqations /.9

30 Starting steps otline of the engineering problem select dependent ariables select coordinate sstem rite don transport eqations rite don bondar conditions ell posed mathematical problem) non-dimensionalie if possible καθ. Γ.Μπεγελές Balance Eqations /.30

31 Eamples of grids and dependent ariables Boiler geometr E/ geometr καθ. Γ.Μπεγελές 4 4 W Κυψέλες Β-Τύπου B Κυψέλες Α-Τύπου S 3 3 Balance Eqations /.31

32 Eamples of grid/ C/d1.6 C/d3.6 θ α κ Μ. Γ. ς έ λ ε γ ε π Balance Eqations /.3

33 Eamples of grid/3 he intake port U alencia) An I.C engine καθ. Γ.Μπεγελές A graphics package is a mst Balance Eqations /.33

34 Methods for grid generation Soltion of a oisson eqation ith Dirichlet or Nemann bondar conditions-forcing fnctions A Laplacian Eqation Method for Nmerical Generation of Bondar-fitted 3D Orthogonal Grids.heodoropolos, G. C. Bergeles Jornal of Comptational hsics, ol. 8,No,1989,pp69-88 Nmerical Grid Generation echniqefor 3D Comple Spaces Glekas, J., Bergeles, G., Athanassiades, N. 3rd Intern. Conference, Comptational Methods and Eperimental measrements, Sept 1986,orto Carras,Springer erlag,ol,pp καθ. Γ.Μπεγελές Balance Eqations /.34

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