SIMULATION ON CONJUGATE HEAT TRANSFER THROUGH SOLID PIPE.

OBJECTIVE:-   (I) TO SETUP SIMULATION ON CONJUGATE HEAT TRANSFER(CHT) IN SOLID PIPE .

                        (II) TO RUN SIMULATION USING GRID DEPENDENCE TEST ON THREE GRIDS .

                       (III) TO OBTAIN VALUE OF Y+ USING POST-PROCESSING .

 

INTRODUCTION :-

THE TERM CONJUGATE HEAT TRANSFER (CHT) IS USED TO DESCRIBE PROCESSES WHICH INVOLVE VARIATIONS OF TEMPERATURE WITHIN SOLIDS AND FLUIDS , DUE TO THERMAL INTERACTION BETWEEN SOLIDS AND FLUIDS. THE EXCHANGE OF THERMAL ENERGY BETWEEN THE TWO PHYSICAL BODIES IS CALLED STUDY OF HEAT TRANSFER, THE RATE OF TRANFERRED HEAT IS DIRECTLY PROPORTIONAL TO THE TEMPERATURE DIFFERENCE THE BODIES . 

FOR EXAMPLE:-

1. HEATING OR COOLING OF A SOLID OBJECT BY THE FLOW OF AIR IN WHICH IT IS IMMERSED.
2. CONDUCTION THROUGH SOLIDS.
3. FREE AND FORCED CONVECTION IN THE GASES/FLUIDS AND THERMAL RADIATION.

CONJUGATE HEAT TRANFER CORRESPONDS WITH THE COMBINATION OF HEAT TRANSFER IN SOLIDS AND HEAT TRANFER IN FLUIDS. IN SOLIDS, CONDUCTION OFTEN DOMINATES WHEREAS IN FLUIDS, CONVECTION USUALLY DOMINATES. EFFICIENTLY HEAT TRANFER IN SOLIDS AND FLUIDS IS THE KEY TO DESIGNING EFFECTIVE COOLERS, HEATERS, OR HEAT EXCHANGERS . FORCED CONVECTION IS THE MOST COMMON WAY TO ACHIEVE HIGH HEAT TRANFER RATE . IN SOME APPLICATIONS , THE PERFORMANCES ARE FURTHER IMPROVED BY COMBINING CONVECTION WITH PHASE CHANGE(FOR EXAMPLE, LIQUID WATER TO VAPOUR PHASE CHANGE).

 

MODES OF HEAT TRANFER:-

• CONDUCTION :- DIFFUSION OF HEAT DUE TO TEMPERATURE GRADIENTS. A MEASURE OF THE AMOUNT OF CONDUCTION FOR A GIVEN GRADIENT IS THE HEAT CONDUCTIVITY .

 

• CONVECTION :- WHEN IS CARRIED AWAY BY MOVING FLUID. THE FLOW CAN BE EITHER CAUSED BY EXTERNAL INFLUENCES , FORCED CONVECTION; OR BOUYANCY FORCES, NATURAL CONVECTION. CONVECTIVE HEAT TRANSFER IS TIGHTLY COUPLED TO THE FLUID FLOW SOLUTION .

 

• RADIATION :- TRANSFER OF ENERGY BY ELECTROMAGNETIC WAVES BETWEEN SURFACES WITH DIFFERENT TEMPERATURES, SEPARATED BY A MEDIUM THAT IS AT LEAST PARTIALLY TRANSPARENT TO THE (INFRARED) RADIATION. RADIATION IS ESPECIALLY IMPORTANT AT HIGH TEMPERATURES, e.g. DURING COMBUSTION PROCESSES .

 

APPLICATION :-

CONJUGATE HEAT TRANFER IS IMPORTANT FOR THE STUDY OF :-

• HEAT TRANSFER FROM ENGINE BLOCK TO THE COOLANT . .
• FORCED CONVECTION COOLING OF ENGINE BLOCK .
• HERT TRANSFER IN GAS EXCHANGER.

                                                                                      

    

 

 

THEORY:-

SOFTWARE USED :-

1. CONVERGE STUDIO:- TO SETUP THE MODEL.      

2. CYGWIN:- FOR RUNNING THE SIMULATION.

3. OPENFOAM:- TO VISUALIZE THE DIFFERENT POST-PROCESSED RESULT.

 

TURBULENCE MODELS :- THESE ARE THOSE MODELS WITH HELP OF WHICH WE ARE ABLE TO RUN SIMULATION IN PHYSICALLY FORM . A TURBULENCE MODEL SHOULD BE CHOSEN IN ACCORDANCE TO FLOW DOMAIN OF THE PIPE .

IN A LAMINAR FLOW , WHEN REYNOLDS NUMBER IS TOO LESS , VISCOUS EFFECTS DOMINATE THE FLOW AND TURBULENCE CAN BE NEGLECTED . THIS FLOW REGIME IS CHARACTERIZED BY REGULAR FLOW LAYERS .

ON THE OTHER HAND , A TURBULENCE FLOW IS CHARACTERIZED BY CHAOTIC AND IRREGULAR PATTERNS THAT ARE ASSOCIATED WITH HIGH REYNOLDS NUMBERS . IN ORDER TO SIMULATE A SUITABLE TURBULENT FLOW , WE SHOULD CHOOSE   AN APPROPRIATE TURBULENCE MODEL .

CURRENTLY, THIS TURBULENCE MODEL ARE SUPPORTED :

• REYNOLDS-AVERAGED NAVIER-STOKES (RANS) 

         a. k-Epsilon

         b. k-Omega-SST

         c. RNG k-omega

• LARGE EDDY SIMULATION (LES) 

        a. SMAGORINSKY

     

 NOTE :- RNG k-`omega` TYPE OF MODELLING IS USED HERE FOR RUNNING SIMULATION .

 

                                       

 

 WHAT IS SUPER-CYCLING IN CONVERGE STUDIO ?

 SUPERCYCLING IS A METHOD USED BY CONVERGE STUDIO IN THE CASE OF CONJUGATE HEAT TRANFER PROBLEMS WITH A SOLID AND FLUID REGION .

 THE INTEGRAL PROBLEM IS THAT BOTH SOLVERS CANNOT RUN AT THE SAME SPEED SINCE SOLVING IN THE FLUID REGION IS MUCH FASTER OWING TO THE TIME SCALE DIFFERENCE FOR HEAT TRANSFER IN FLUID AS COMPARE TO THE SOLIDS .   THIS CAUSES PROBLEMS   DURING THE SOLUTION GIVEN THAT THE SOLID SIDE SOLVER WOULD NOT HAVE REACHED A STEADY/CONVERGENCE IN THE TIME FLUID SOLVER DOES . AND THAT IS WHERE THE CONCEPT OF SUPERCYCLING COMES .

 THE BASIC IDEA OF SUPERCYCLING IS THAT THE SOLVER FOR FLUID REGION IS PAUSED UNTIL THE SOLVER FOR THE SOLID DOMAIN CONVERGES .THIS PAUSING IS DONE IN INTERVALS THAT CAN BE SET BY THE USER .

 

                                                    

                                                                                                                 

 

SUPERCYCLING METHOD SETUP PROCEDURE :-

 

1.The first option is to set the initial time at which the super-cycle solver begins storing data. This can be set from 0 to any value less than or equal to the total simulation time.

 

2.The solver used for the solid region can be a steady or transient solver that is independent of the solver set in the run parameters window and can be steady or transient.

 

3.The time length for each cycle determines the intervals at which the fluid domain is paused and the solid domain solved for. This interval is taken into account after the initial pause of the fluid solver at 1.

 

4.The CFL number, SIE tolerance and relaxation factors are solver parameters that are given for the purpose of accuracy and speed of the solver.

 

5.An output point must be defined in the solid region to let the solver know at which point the temperature data must be calculated. This point must be given in the solid domain as the solver runs for the solid domain and once this value converges, the fluid domain solver is continued.

 

(A) TYPE OF MODEL:-

IT IS BASICALLY A SOLID PIPE WHICH HAS SOME THICKNESS WITH CROSS-SECTION OF CIRCULAR PROFILE  .

 

                                          

 

 

(B) DIMENSION OF CHANNEL:-

      1. DIAMETER OF ELBOW =0.06 m

      2. THICKNESS OF PIPE = 0.005 m 

      3. LENGTH OF ELBOW BODY= 0.25 m

 

                                         

 

(D) INITIAL CONDITION :-

       BASICALLY THE PIPE IS DIVIDED INTO TWO REGIONS:- SOLID REGION AND FLUID REGION . HERE WE HAVE CONSIDER FOLLOWING THINGS FOR PIPE IN CONVERGE STUDIO:-

       1. TYPE OF FLUID USED:- GAS (MIXTURE OF BASICALLY OXYGEN AND NITROGEN).

       THE FLUID MATERIAL PROPERTIES AND BEHAVIOUR ARE DEFINED BY THERMAL FLUID MODEL .

       2. TYPE OF SOLID USED:- ALUMINIUM.

      THE SOLID MATERIAL PROPERTIES AND BEHAVIOUR ARE DEFINED BY THERMAL SOLID MODEL .

 

STREAMS :-

A STREAM IS A COLLECTION OF CONNECTED REGIONS OF THE SAME PHASE (FLUID , SOLID , ETC.)

CONVERGE SOLVES FOR EACH STREAM INDEPENDENTLY EXCEPT FOR COUPLED INTERFACES .

YOU MUST ASSIGN EACH REGION TO A STREAM . IT IS RECOMMENDED THAT 

• STREAM ID=0 FOR FLUIDS .
• STREAM ID =1 FOR STATIC SOLIDS. 
• STREAM ID >1 FOR MOVING SOLIDS.

NOTE:- IF TWO SOLIDS HAVE SAME MOTION PROFILE , ASSIGN THEM TO THE SAME STREAM . OTHERWISE , ASSIGN THEM DIFFERENT STREAM ID .

 

                                                      

 

 

                                                      

 

(E)  BOUNDARY CONDITION :-

FOR PRESSURE :-

AT INLET, NO PREDEFINED VALUE (NEUMANN CONDITION).

AT OUTLET, 101325 PASCAL (DRICHLET CONDITION).

FOR VELOCITY :-

AT INLET, 3.64 m/s (DRICHLET CONDITION).

AT OUTLET, NO PREDEFINED VALUE(NEUMANN CONDITION). 

NOTE:-

HERE VELOCITY IS DEFINED BY :-

Re = (rho*V*D)/(Dynamic viscosity)   , where Re=7000 .  

 

                                                        

 

INTERFACE :- THIS BOUNDARY SEPARATES THE TWO TYPES OF MATERIALS OR PHASES .

INTERFACE:COUPLING

• CONTACT RESISTANCE=0 : TEMPERATURE CONTINUITY AT THE INTERFACE .     
• CONTACT RESISTANCE>0 : TEMPERATURE DISCONTINUITY AT THE INTERFACE .    

WALL :- THIS BOUNDARY CONTAINS THE SOLID PORTION NOT IN CONTACT WITH THE SECONDARY MATERIAL OR PHASE .

FOR A SOLID WALL , POSSIBLE TEMPERATURE BOUNDARY CONDITION ARE:-

• DIRICHLET : SPECIFY THE WALL TEMPERATURE DIRECTLY .
• HEAT FLUX : SPECIFY HEAT FLUX PER SURFACE AREA FOR A BOUNDARY .

•  A POSITIVE VALUE SPECIFIES ENERGY EXITING THE SOLID .
•  A NEGATIVE VALUE SPECIFIES ENERGY ENTERING THE SOLID .

• CONVECTION :- SPECIFY A HEAT TRANFER COEFFICIENT (HTC) AND A FAR FIELD TEMPERATURE .

NOTE : HERE IN THE SETUP , WE HAVE USED HEAT FLUX BOUNDARY CONDITION WHICH HAVE VALUE OF 1000 .

 

(F) CRITERIA FOR RUNNING SIMULATION :- 

       1. INTIAL TIME STEP :- 1E-6 SECOND .

       2. TIME PERIOD FOR SIMULATION :- 0.5 SECONDS .

       3. NUMBER OF CYCLES FOR SIMULATION:- 0.01 

 

                                                                               

 

 

(G)  GRID SIZE : -

         dx=4mm ; dy=4mm ; dz=4mm .  

                                                                               

 

 

(H)  SOLVER TYPE USED:-  DENSITY BASED TRANSIENT SIMULATION .

      

                            

 

 

PRE-PROCESSING STEPS :-

AFTER SETTING UP THE MODEL, WE SENT OUT DATA TO A PARTICULAR LOCATION WHERE ALL THE DETAIL OF CHANNEL IS AVAILABLE REQUIRED BY FURTHER SOFTWARE FOR SIMULATION.

THEN AFTER GENERATING AND STORING THE DATA, WE USED CYGWIN SOFTWARE WHOSE WORK IS TO RUN SIMULATION. 

PROCEDURE FOR RUNNING SIMULATION:-

1. OPEN "CYGWIN" IN COMMAND PROMPT.

2. THEN TRACE THE LOCATION WHERE DATA IS STORED FROM CONVERGE STUDIO.

3. NOW TYPE "converge.exe" FOR SERIES SIMULATION OR "mpiexec.exe -n 4 converge" FOR PARALLEL SIMULATION IN THAT FILE LOCATION.

4. AFTER DOING THIS, SIMULATION WILL START WORKING AND AND WILL STOP AFTER GIVEN NUMBER OF CYCLES (BOTH IMPLICITLY AND EXPLICITLY DEPANDING UPON SIMULATION PERIOD).

5. THIS IS HOW THE WORK OF CYGWIN COMPLETES. AFTER THIS USE POST_CONVERT COMMAND IN OUTPUT FILE IS USED WHICH IS GENERATED BY CYGWIN. IT WILL CONVERT THE GENERATED OUT FILE FROM CYGWIN INTO A COMPATIBLE FROM WHICH WILL BE EASILY ACCESSED BY THE PARAFOAM.

    NOW WE OPEN THE PARAFOAM SOFTWARE FOR POST PROCESSING TO VISUALIZE THE RESULT AND COMPARE IT.

    

CONJUGATE HEAT TRANSFER MODELING SETUP :-

• THERE ARE SEVERAL STEPS TO SETUP A CONJUGATE HEAT TRANSFER SIMULATION .

• DEFINE FLUID AND SOLID MATERIALSN AND THEIR PROPERTIES .
• CONFIGURE THE FLUID AND SOLID REGIONS .
• ASSIGN REGIONS TO STREAMS .
• USE APPROPRIATE BOUNDARY CONDITIONS FOR THE BOUNDARIES THAT SEPARATE DIFFERENT REGIONS .
• SELECT TIME CONTROL METHOD (TRANSIENT , STEADY-STATE , OR SUPER-CYCLE) .
• SET UP POST PROCESSING .

 

POST-PROCESSING STEPS :-

1. IN OPENFOAM, CLICK ON OPEN FROM FILE MENU FROM DROP-DOWN MENU WHICH IS PRESENT AT TOP-LEFT OF THE SCREEN.

2. SEARCH FOR FILE WHERE FILE IS GENERATED BY POST CONVERT COMMAND AND OPEN IT. REMEMBER IT MUST HAVE ".vtm" DOMAIN AFTER FILE NAME.

3. CLICK ON APPLY, GEOMETRY WILL ORIGINATE. AFTER THIS CLICK ON SLICE OPTION TO SECTION THE GEOMETRY . NOW CLICK ON DROP DOWN MENU (SURFACE WITH EDGES) FROM TOP TOOLBAR OPTIONS . THIS WAY THE MESH WILL GENERATE .

4. AFTER IT, CLICK ON LINE PROBE OPTION TO SEE THE DIFFERNT PHYSICAL QUANTITES AND APPLY IT IN ACCORDANCE TO YOUR CHOICE .

5. IN THIS WAY, THE GRAPH WILL GENERATE AND WE WILL SEE DIFFERENT PARAMETERS ACCORDING TO THAT APLLIED LINE PLOT.

6. IN THIS WAY, OUR POST-PROCESSING IS DONE.

 

POSTPROCESSING RESULTS :-

MESH GENERATION WITH SOLID AND FLUID REGION :-

                                                                       

 

                                                                      

 

HERE SOLID AND FLUID REGION IS DENOTED BY "RED SHADED REGION" AND "BLUE SHADED REGION" RESPECTIVELY .

 

VELOCITY AND PRESSURE PROFILE :-

FOR SUPERCYCLE STAGE INTERVAL = 0.01 sec

AT GRID SIZE=0.004 m

    

 

  

 

 

AT GRID SIZE=0.003 m

 

  

 

  

   

AT GRID SIZE=0.002 m

 

 

  

 

CONTOUR PLOT FOR Y+ IN THE PIPE :-

AT GRID =0.002 FOR SUPERCYCLE INTERVAL 0.03

 

 

AT GRID =0.003 FOR SUPERCYCLE INTERVAL 0.03

 

 

 

AT GRID =0.004 FOR SUPERCYCLE INTERVAL 0.03

 

 

               

ANIMATION OF TEMPERATURE VARIATION IN PIPE FOR DIFFERENT GRID SIZE :-

AT GRID SIZE=0.004 m

 

 

AT GRID SIZE=0.003 m

 

 

AT GRID SIZE=0.002 m

 

 

OBSERVATION WITH EXPLAINATION :-

IN THE PRESSURE CONTOUR PLOT, WE OBSERVED THAT PRESSURE HAVE DECREASED TO OUTLET IN COMPARSION TO THE INLET SECTION . THIS IS BECAUSE WE HAVE ALREADY DEFINED PREDEFINE VALUE IN DOMAIN AND AT OUTLET SECTION DUE TO WHICH WE ARE SEEING HIGHER VALUE AT INLET JUST DUE TO FLUID MOTION AS PRESSURE MOVES FROM HIGHER TO LOWER PRESSURIC REGION.

IN THE VELOCITY CONTOUR PLOT, WE OBSEVED THAT VELOCITY HAVE INCREASE HIGHLY AT OUTLET IN COMPARISON TO THE INLET SECTION . THIS IS BECAUSE VELOCITY AND PRESSURE IS INVERSELY PROPORTIONAL TO EACH OTHER IN ACCORADANCE TO BERNOUILLE PRINCIPLE DUE TO WHICH VELOCITY IS HIGHER AT LOW PRESSURE REGION AND LOW AT HIGHER PRESSURIC REGION . 

APART FROM THIS , WE SEE THAT A HYDRODYNAMIC BOUNDARY LAYER IS DEVELOPING AT THE INTERFACE(SURFACE) OF SOLID AND FLUID REGION WHICH CAUSES REDUCTION IN VELOCITY OF FLUID . THIS HYDRODYANMIC BOUNDARY LAYER  DECREASES AS THE VELOCITY INCREASES AND LEADS TO INCREASE IN FREESTREAM VELOCITY BY MERGING TO THE SURFACE OF FLUID ADJACENT TO SOLID REGION . THIS HYDRODYANMIC BOUNDARY LAYER IS DEVELOPED DUE TO PRESENCE OF SHEAR STRESS AT THE SURFACE OF PIPE . AND SINCE WE HAVE PROVIDED NO SLIP CONDITION AT SURFACE OF FLUID , IT IS OBVIOUS THAT THERE WILL BE SHEAR STRESS AT THE SURFACE OF FLUID. SO, AS THE SHEAR STRESS INCREASES FROM THE SURFACE OF FLUID, IT DIRECTLY LEADS TO AN INCREASE IN HYDRODYANMIC FLUID LAYER.

IN TEMPERATURE ANIMATION, WE NOTICED THAT TEMPERATURE HAVE INCREASED FROM HIGHER TO LOWER VALUE IN FLUID DOMIAN .WHILE IN SOLID DOMAIN, TEMPERATURE HAVE REMAIN CONSTANT OVERALL ACROSS THE SOLID REGION .THIS IS BECAUSE HEAT TRANSFER RATE IS CONSTANT IN SOLID REGION DUE TO WHICH THERE IS NO IMPACT ON TEMPERATURE. WHILE AS IN FLUID REGION, TEMPERATURE IS DOMINATED BY FLUID FLOW WHERE TEMPERATURE DECREASES AS MOTION OF FLUID(VELOCITY) INCREASES.

TEMPERATURE INCREASES AT THE BOUNDARY PLATE SURFACE (SURFACE BETWEEN SOLID AND FLUID REGION) AS WE MOVE FORWARD TOWARD THE FLUID DOMAIN OF PIPE. THIS IS DUE TO DEVELOPMENT OF THERMAL BOUNDARY LAYER AT THE SURFACE OF SOLID REGION ADJACENT TO THE FLUID REGION. THIS DEVELOPED THERMAL BOUNDARY LAYER GENERATES AN INCREASE IN TEMPERATURE AT THE SURFACE BOUNDARY OF PIPE DUE TO WHICH WE ARE VIEWING HIGHER TEMPERATURE AT THE SURFACE OF SOLID REGION BOUNDARY. HERE IF THE VELOCITY OF FLUID INCREASE MORE THEN WE WILL SEE A MORE INCREASE IN TEMPERATURE VALUE AT THE SURFACE.

 

HYDRODYANMIC BOUNDARY LAYER :- BOUNDARY LAYER UPTO TO WHICH SHEAR STRESS CAN BE FEELED BY THE MOVING FLUID . THIS BOUNDARY LAYER IS RESPONSIBLE FOR DECREASE IN FREESTREAM VELOCITY.

THERMAL BOUNDARY LAYER :- BOUNDARY LAYER UPTO TO WHICH THERMAL STRESS CAN BE FEELED BY THE MOVING FLUID . THIS BOUNDARY LAYER IS RESPONSIBLE FOR INCREASE IN TEMPERATURE OF MOVING FLUID.

SLIP CONDITION :- THAT CONDITION IN WHICH VELOCITY OF FLUID IS ZERO AT THE SURFACE RELATIVE TO BOUNDARY.

NO-SLIP CONDITION :- THAT CONDITION IN WHICH VELOCITY OF FLUID IS NON-ZERO AT THE SURFACE RELATIVE TO BOUNDARY.

IN TERMS OF CONVERGENCE , WE CONCLUDED THAT SUPERCYCLE TECHINIQUE PLAYS AN IMPORTANT IN REACHING CONVERGENCE WHERE WE UNDERSTAND THAT AS WE INCREASE THE TIME INTERVAL , WE WILL INCREASE THE TIME CONSUMPTION FOR REACHING CONVERGENCE ALTHOUGH THERE MIGHT BE NO DIFFERENCE IN CONVERGENCE AND IN CONVERGENCE RATE . 

 

IN CASE OF SUPERCYCLING ,WE OBSERVE THAT RATE OF CONVERGENCE IS ALTHOUGH SAME IN ALL THE SUPERCYCLING TIME INTERVAL AS DEPICTED IN PLOTS BELOW . BUT, WE NOTICED THAT TIME CONSUMED IN REACHING THIS CONVERGENCE IS DIFFERENT IN ALL THE CASES i.e FOR SUPERCYCLE 0.01 , 0.02 , 0.03 . WHERE AS WE INCREASE THE TIME INTERVAL , WE NOTICED THAT TIME CONSUMPTION ALSO INCREASED.                                                                                           

THE TIME CONSUMPTION FOR CONVERGENCE IN TERMS OF SUPERCYCLE INTERVAL WAS IN THE FLOLLOWING WAY :-

                                                                                                                                                                          0.03>0.02>0.01

                                

 

                                       

FOR SUPERCYCLE INTERVAL=0.01                 

Program used 3862.797528 seconds.

Summary of total time for:
load balance = 0.76 seconds ( 0.02%)
solving transport equations = 3384.85 seconds (87.63%)
move surface and update grid = 8.28 seconds ( 0.21%)
update boundary conditions = 136.16 seconds ( 3.52%)
combustion = 0.00 seconds ( 0.00%)
spray = 0.01 seconds ( 0.00%)
writing output files = 262.70 seconds ( 6.80%)

                                     

FOR SUPERCYCLE INTERVAL=0.02                 

Program used 5047.827863 seconds.

Summary of total time for:
load balance = 1.03 seconds ( 0.02%)
solving transport equations = 4539.80 seconds (89.94%)
move surface and update grid = 7.97 seconds ( 0.16%)
update boundary conditions = 148.21 seconds ( 2.94%)
combustion = 0.00 seconds ( 0.00%)
spray = 0.01 seconds ( 0.00%)
writing output files = 276.61 seconds ( 5.48%)

                                      

FOR SUPERCYCLE INTERVAL=0.03

Program used 5693.058302 seconds.

Summary of total time for:
load balance = 0.79 seconds ( 0.01%)
solving transport equations = 5134.63 seconds (90.19%)
move surface and update grid = 9.24 seconds ( 0.16%)
update boundary conditions = 161.28 seconds ( 2.83%)
combustion = 0.00 seconds ( 0.00%)
spray = 0.01 seconds ( 0.00%)
writing output files = 308.86 seconds ( 5.43%)

 

 

WE OBSERVED THAT Y+ INCREASES AS WE INCREASE THE GRID SIZE i.e THE LAYER OF Y+ GOES ON INCREASING AS THE GRID BECOME COARSE. THIS IS PROVED BY THE SUBSTANIAL EVIDENCE GIVEN BELOW :-

 

FOR GRID=0.004                                                                                                                                                                                                     

 

             

 

FOR GRID=0.003 

 

               

 

FOR GRID=0.002

 

    

 

HERE WE CAN EASILY SEE ABOVE THAT HOW Y+ NUMBER IS VARYING ON ACCOUNT OF CHANGE IN SIZE OF GRID SIZE IN INCREASING ORDER. THE REASON FOR CHANGE IN THIS Y+NUMBER IS THAT AS THE GRID BECOME COARSE , IT LEADS TO AN INCREASE IN THE Y+NUMBER.

Y+ NUMBER :- IT IS A NON-DIMENSIONAL TERM (IN BOUNDARY LAYER THEORY OF FLUID DYNAMICS) THAT IS USED TO UNDERSTAND HOW COARSE OF FINE OUR GRID IS. 

SIGNIFICANCE OF Y+ NUMBER :-IT HELPS US IN KNOWING WHAT KIND OF WALL FUNCTION , WE NEED TO USE IN ORDER FOR SOLVER TO REACH THE SOLUTION AT THE WALL OF FLUID . THESE WALL FUNCTIONS HELPS US IN PREDICTING THE RESULT AT THE WALL ACCURATELY EVEN AT COARSE GRID .   

WALL FUNCTION LIKE NO SLIP CONDITION CAN BE USED TO REACH THE ACCURATE RESULT AT WALL . BUT HERE IS A PROBLEM , THIS CONDITION CAN USED ONCE IF WE KNOW THAT FLOW IS COMPLETELY LAMINAR WHICH BECOMES LIABILITY FOR US . SO IT BECOMES NECESSARY FOR US TO HAVE A CONVENIENT TERM WHICH CAN HELP US IN KNOWING THE CONDITION AT THE WALL AND REACH THE ACCURATE RESULT AT THE WALL OF FLUID . ONCE WE KNOW THAT WHAT'S CONDITION IS ON WALL , WE CAN DEFINETLY REACH OUT FOR AN ACCUARATE SOLUTION FOR FLUID IN WHOLE DOMAIN.    

                                                                                                                                                                                                                                                                   

CONCLUSION :-

WE CONCLUDED THAT AS THE VELOCITY OF FLUID INCREASES , IT LEADS TO DECREASE IN HYDRODYANMIC BOUNDARY LAYER DUE TO WHICH THE RATE OF DIFFUSIVITY OF MOMENTUM IS LESS IN THE FLUID CAUSES MERGER OF FREESTREAM VELOCITY TO THE SURFACE OF THE FLUID. SO, WE UNDERSTAND THAT AS VELOCITY INCREASES , IT LEADS TO DECREASE IN SHEAR STRESS.

IN SOLID REGION , THE TEMPERATURE IS CONSTANT THROUGHOUT AS THE RATE OF HEAT TRANSFER IS SAME THROUGHOUT OVERALL PERIOD OF TIME . IN FLUID REGION , AS THE VELOCITY OF FLUID INCREASES , IT CAUSES AN DECREASE IN TEMPERATURE AS RATE OF THERMAL DIFFUSIVITY IS MORE IN THE DOMAIN DUE TO WHICH NET HEAT TRANSFER IS LESS IN FLUID DOMAIN HENCE TEMPERATURE DECREASES .

HERE AT THE SURFACE OF FLUID IN CASE OF TEMPERATURE , THE THERMAL BOUNDARY LAYER DEVELOPS AT THE SURFACE WHICH CAUSES AN EXTRA INCREASE IN VALUE OF TEMPERATURE AS RATE OF THERMAL DIFFUSIVITY IS LESS DUE TO WHICH THE HEAT TRANSFER FROM SOLID REGION IS MORE WHEN VELOCITY OF FLUID IS MORE . HENCE CAUSING AN INCREASE IN TEMPERATURE BOUNDARY LAYER AT THE SURFACE OF FLUID WHICH WE CAN EASILY SEE IN THE ANIMATION . SO WE CONCLUDED THAT AS VELOCITY WILL INCREASE , THERMAL BOUNDARY LAYER WILL INCREASE HENCE WILL LEAD TO RISE IN TEMPERATURE AT THE SURFACE .

SO HERE IN PLOT , WE UNDERSTAND THAT IF WE HAVE TO USE NO SLIP CONDITION AT THE WALL AND OBTAIN ACCURATE RESULT . SO IN THAT CASE , WE HAVE TO USE FINER GRID SIZE FOR NO SLIP WHICH IS PROVIDED SUITABLY BY THE GRID SIZE OF 0.002 m . ALTHOUGH , THE Y+ NUMBER COMES UNDER THE DOMAIN OF TRANSITION REGION , NOT VISCOUS-SUB LAYER STILL THE RESULT WILL BE ACCURATE SOMEHOW AT THE WALL . BUT IF WE HAVE TO ACHIEVE A BEST RESULT AT THE WALL , THEN WE HAVE TO DECREASE THE GRID SIZE IN ORDER TO OBTAIN Y+ NUMBER LESS THAN 10 AND MAKE CONDITION SUITABLE FOR VISCOUS SUB LAYER AND A GET A PROPER RESULT .