SIMULATION ON PRANDTL MEYER SHOCK PROBLEM AND ANALYSING THE EXPANSION WAVE.

THEORY:-

SHOCK WAVE :- IN PHYSICS , A SHOCK WAVE IS A TYPE OF PROPAGATING ADRUPTNESS OR DISTURBANCE THAT MOVES FASTER THAN THE LOCAL SPEED OF SOUND IN THE MEDIUM . LIKE AN ORDINARY WAVE , A SHOCK WAVES ALSO CARRIES ENERGY AND PROPAGATES THROUGH A MEDIUM BUT HAVE DISCONTINUITY IN IT\'S PHYSICAL PROPERTIES LIKE  PRESSURE , TEMPERATURE AND DENSITY .

SHOCK WAVES ARE HIGHLY IRREVERSIBLE PROCESSES .

TYPES OF SHOCK FLOW :-

 1. NORMAL :- At 90° (perpendicular) to the shock medium\'s flow direction.

 2. OBLIQUE :- At an angle to the direction of flow.

 3. BOW :- Occurs upstream of the front (bow) of a blunt object when the upstream flow velocity exceeds Mach 1.

 

PRANDTL-MEYER EXPANSION FAN :-

A SUPERSONIC EXPANSION FAN, TECHNICALLY KNOWN AS PRANDTL–MEYER EXPANSION FAN, A TWO-DIMENSIONAL SIMPLE WAVE, IS A CENTERED EXPANSION PROCESS THAT OCCURS WHEN A SUPERSONIC FLOW TURNS AROUND A CONVEX CORNER. THE FAN CONSISTS OF AN INFINITE NUMBER OF MACH WAVES, DIVERGING FROM A SHARP CORNER. WHEN A FLOW TURNS AROUND A SMOOTH AND CIRCULAR CORNER, THESE WAVES CAN BE EXTENDED BACKWARDS TO MEET AT A POINT.

EACH WAVE IN THE EXPANSION FAN TURNS THE FLOW GRADUALLY (IN SMALL STEPS). IT IS PHYSICALLY IMPOSSIBLE FOR THE FLOW TO TURN THROUGH A SINGLE "SHOCK" WAVE BECAUSE THIS WOULD VIOLATE THE SECOND LAW OF THERMODYANMICS.

ACROSS THE EXPANSION FAN, THE FLOW ACCELERATES (VELOCITY INCREASES) AND THE MACH NUMBER INCREASES, WHILE THE STATIC PRESSURE, TEMPERATURE AND DENSITY DECREASE. SINCE THE PROCESS IS ISENTROPIC, THE STAGNATION PROPERTIES (E.G. THE TOTAL PRESSURE AND TOTAL TEMPERATURE) REMAIN CONSTANT ACROSS THE FAN.

 

BOUNDARY CONDITION :- IN PHYSICS ,THESE ARE THOSE CONDITIONS WHICH ARE APPLIED AT THE BOUNDARY OF THE DOMAIN . IT SPECIFIES THE CONDITION AT BOUNDARY ON THE BASIS OF WHAT THE GOVERNING EQUATION DEFINES THE CONDITION OF DOMAIN ALONG WITH THE HELP OF INTIAL CONDITIONS .

TYPE OF BOUNDARY CONDITION :-

1. DRICHLET CONDITION :- THOSE CONDITION WHERE THE ACTUAL VALUE OF PHYSICAL QUANTITY IS SIMPLY APPLIED AT THE BOUNDARY  .

    FOR EXAMPLE ,

                                        `F(x)=T(x)`

  here function defines the value at boundary in the form of temperature varying with distance .

2. NEUMANN CONDITION :- THOSE CONDITION WHERE THE ACTUAL VALUE IS SIMPLY APPLIED AT THE BOUNDARY IN THE FORM OF TRUE SOLUTION OF ORDINARY OR PARTIAL DIFFERNTIAL EQUATION .

    FOR EXAMPLE ,

                                         `F(x)=((D^2T)/(D^2x))_(atx=0&L .`

HERE VALUE OF PHYSICAL QUANITY IS DEFINED AT \'0\' AND \'L\' m IN THE FORM OF TRUE SOLUTION OF DIFFERENTIAL EQUATION. 

ALONG WITH THIS THERE ARE OTHER TYPE OF BOUNDARY CONDITION LIKE CAUCHY , ROBIN AND MIXED ONE BUT ABOVE TWO ARE PREDOMINANT ONE IN COMPUTATIONAL FLUID DYNAMICS .

 

SOFTWARE USED:-

1. CONVERGE STUDIO:- TO SETUP THE MODEL.

2. CYGWIN:- FOR RUNNING THE SIMULATION.

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

 

PRE-PROCESSING STEPS:-

HERE WE HAVE CONSIDER FOLLOWING THINGS FOR CHANNEL IN CONVERGE STUDIO:-

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

 

(B) TYPE OF MODEL:-

• OPEN FROM BOTH SIDE.
• WALL FROM BOTTOM AND TOP.
• 2D SURFACE FROM FRONT AND BACK.

 

                                                      

 

(C) DIMENSION OF MODEL:-

      1. LENGTH=65 m

      2. BREADTH= 45.15 m

      3. HEIGHT=8.15 m

 

                                                     

 

(D) SIZE OF GRID USED:-1 m FOR LENGTH, HEIGHT AND BREADTH.

       HERE ADAPTIVE MESH REFINEMENT IS USED WHERE SGS=0.1, 0.05, 0.02 .

 

                                                                       

 

 

                                                  

 

(E) BOUNDARY CONDITION :-

 1.  FOR SUPERSONIC CASE :-

      VELOCITY AT INLET=680 m/s. (DRICHLET)

      VELOCITY AT OUTLET= unknown. (NEUMANN)

      PRESSURE AT INLET AND OUTLET=unknown.(NEUMANN)

 

                                    

 

 2.  FOR SUBSONIC CASE :-

      VELOCITY AT INLET=100 m/s. (DRICHLET)

      VELOCITY AT OUTLET= unknown. (NEUMANN)

      PRESSURE AT INLET AND OUTLET=unknown.(NEUMANN)

 

                                   

 

(F) INITIAL CONDITION :-

       VELOCITY (IN DOMAIN) :- 100m/s.

 

                                                              

 

(G) TIME STEP FOR RUNNING SIMULATION:- 25000 CYCLES.

                                                                    

 

(H) SOLVER TYPE USED:- DENSITY BASED STEADY SOLVER .

 

                                           

 

  

 

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.

4. AFTER DOING THIS, SIMULATION WILL START WORKING AND AND WILL STOP AT GIVEN TIME STEP.

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 TO VISUALIZE THE RESULT AND COMPARE IT.

 

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 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.

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

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

 

BASE MESH OF GEOMETRY (WITH NO ADAPTIVE MESH REFINEMENT) :-

 

                                                   

 

                                                                      

BASE MESH OF GEOMETRY (WITH ADAPTIVE MESH REFINEMENT) :-

 

                                                  

 

 

                                                                  

 

A.) SUPERSONIC CASE :-

PRESSURE CONTOUR WITH ADAPTIVE MESH REFINEMENT :-

AT SGS= 0.1 

 

                                                

 

                                                                      

 

 AT SGS = 0.05 

 

                                              

 

 

                                                                      

 

 AT SGS = 0.02

 

                                           

                                                                     

 

 

VELOCITY CONTOUR WITH ADAPTIVE MESH REFINEMENT :-

 

 AT SGS = 0.1 

 

                                          

                                                                    

 

 AT SGS= 0.05 

 

                                        

                                                                   

 

 AT SGS =0.02

 

                                         

                                                                  

B.) SUBSONIC CASE :-

 PRESSURE CONTOUR WITH ADAPTIVE MESH REFINEMENT :-

 AT SGS= 0.1 

 

                                            

                                                                  

 

 AT SGS= 0.05

 

                                        

       

                                                            

 

 AT SGS= 0.02

 

                                        

                                                          

 

 VELOCITY CONTOUR WITH ADAPTIVE MESH REFINEMENT :-

 AT SGS=0.1

 

                                        

                                                         

 

AT SGS=0.05

 

                                       

 

                                                           

 

 AT SGS=0.02

 

                                       

                                                           

 

              

 

 

 ANIMATION OF TEMPERATURE SHOCK FLOW PROFILE (SUBSONIC CASE) :-  

 AT SGS =0.1

 

   

         

AT SGS =0.05

 

 

         

AT SGS =0.02

 

 

 

 ANIMATION OF TEMPERATURE SHOCK FLOW PROFILE (SUPERSONIC CASE) :-  

 

 AT SGS =0.1

 

             

 

 

 AT SGS =0.05

 

          

           

 AT SGS =0.02

 

         

 

 

 OBSERVATION WITH REASON :-

1. SUBSONIC CASE:- 

HERE IN THIS CASE , WE OBSERVE THAT FLOW WHICH IS FLOWING AT THE SPEED OF 100 m/s GETS EXTINGUISHES AS WE START THE SIMULATION  IN TERMS OF VELOCITY . THE REASON BEHIND IT SIMPLE AS WE KNOW AIR FLOWS AT SPEED OF 343 m/s IN ATMOSPHERIC CONDITION .SO AS WE START SIMULATION , THE PREDEFINED AIR IN DOMAIN BLOWS AWAY AND WE GET NEW FRESH AIR WHICH FLOW AT NORMAL SPEED .

WHEREAS , IF WE CONSIDER THE FLOW IN TERMS OF TEMPERATURE , WE OBSERVE THAT TEMPERATURE IS SLIGHTLY INCREASED FROM THE PREDEFINED TEMPERATURE OF 280 DEGREE CELCIUS . THIS IS BECAUSE WHEN AIR MOVES WITH HIGHER VELOCITY , THE MOLECULES IN THE AIR GETS ENERGIES DUE TO INTER-MOLECULAR VIBRATION . HENCE AS A RESULT OF WHICH TEMPERATURE INCREASES DIRECTLY .

IN TERMS OF PRESSURE , WE OBSERVED THAT PRESSURE IS HIGHER AT OUTLET SECTION IN COMPARISON TO INLET SECTION . THIS IS BECAUSE PRESSURE ALWAYS INCREASES IN DUCT PROFILE IN TERMS OF SUBSONIC CASE .

AS IN SUBSONIC CASE FOR DUCT PROFILE ,

                                                                   FOR dA>0 , dV<0 and dP>0.

WHEREAS IN CASE FOR NOZZLE PROFILE ,

                                                                   FOR dA<0 , dV>0 and dP<0.

EFFECT OF SGS ON SHOCK LOCATION(SUBSONIC CASE) :-

THE SGS GIVES US MORE PRECISE DETAIL INFORMATION OF VARIATION OF PHYSICAL QUANTITY SPECIALLY IN MORE CRITICAL REGION WHERE IT IS HAVING MUCH WIDE VARIATION . THE MORE PRECISE SGS HELPS US IN GETTING MORE ACCURATE RESULT AS WE SAW ABOVE IN THE DIFFERENT CASES OF SGS. APART FROM IT , IN SUBSONIC CASE THERE IS NO EFFECT OF SGS IN THIS CASE AS HERE FLOW HAS NOT ATTAIN THE CRITICAL POINT OF SUPERSONIC MACH NUMBER DUE TO WHICH WE CAN\'T OBSERVE SHOCK IN FLOW HENCE WE CAN\'T SEE EFFECT OF SGS AS ALL PHYSICAL QUANTITY IS IN NORMAL CONDITION .

 

2. SUPERSONIC CASE :-

HERE IN THIS CASE , WE HAVE PREDEFINED THE SPEED OF 680 m/s WHICH IS APPROXIMATELY ABOUT 2 MACH NUMBER . HERE AS WE START THE SIMULATION , WE OBSERVE THAT THERE IS SUDDEN SHOCK IN FLOW . THE SUDDEN ORIGINATION OF SHOCK IS DUE TO SUDDEN INCREASE IN AREA OF PROFILE (DUCT) WHERE SUDDEN CHANGE IN AREA CAUSES THE CASE OF EXPANSION FAN .

HERE WE OBSERVED THAT TEMPERATURE ALSO INCREASES IN THIS CASE , THE REASON IS AGAIN SAME THAT WHEN FLUID MOVES WITH HIGHER VELOCITY , IT TENDS TO GENERATE FRICTION BETWEEN AIR MOLECULES DUE TO WHICH THE RELATIVE TEMPERATURE OF FLOW INCREASES .

IN TERMS OF PRESSURE , WE OBSERVED THAT PRESSURE IS LOWER AT INLET SECTION IN COMPARISON TO OUTLET SECTION . THIS IS BECAUSE PRESSURE DECREASES IN DUCT PROFILE IN CASE OF SUPERSONIC CASE .

IN CASE FOR DUCT PROFILE IN SUPERSONIC ,

                                                                   FOR dA>0 , dV>0 and dP<0.

WHEREAS IN CASE FOR NOZZLE PROFILE ,

                                                                   FOR dA<0 , dV<0 and dP>0.

 

ALSO,

                                                                                                                                                                                        

 

PRANDTL MEYER EXPANSION FAN (REASON FOR PHYSICS BEHIND SHOCK WAVE) :-

FOR THE PURPOSE OF COMPARISON IN SUPERSONIC FLOW , EXPANSION FAN (ALSO KNOWN AS PRANDTL-MEYER EXPANSION FAN) IS USED WHO\'S WORK IS TO INCREASE THE EXPANSION ADDITIONALY . THE EXPANSION WAVE CAUSE COLLISION AND RECOMBINATION WITH THE SHOCK WAVE WHICH MAKES A PROCESS OF DISTRUCTIVE INTERFERENCE .

THE ENERGY AND SPEED OF A SHOCK WAVE DISSIPATES RELATIVELY QUICKLY WITH THE DISTANCE. WHEN A SHOCK WAVE PASSES THROUGH MATTER , IT ENERGY PRESERVES BUT THE ENTROPY INCREASES WHERE CHANGE IN MATTER PROPERTIES LEADS TO DECREASE IN ENERGY WHICH CAN BE EXTRACTED AS A USEFUL WORK AND A DRAG FORCE ON SUPERSONIC FLOW OR OBJECTS . 

WHEN AN OBJECT MOVES FASTER THAN THE INFORMATION CAN PROPAGATE INTO SUROUNDING FLUID , THEN FLUID NEAR THE DISTURBANCE OR SHOCK CANNOT REACT EASILY BEFORE DISTURBANCE ARRIVES .

SHOCK WAVES ARE FORMED WHEN PRESSURE FRONT MOVES AT SUPERSONIC SPEED AND PUSHES ON SURROUNDING AIR . AT THE REGION , WHERE THIS OCCUR SOUNDS WAVE TRAVELLING AGAINST THE FLOW REACHES A POINT WHERE THEY CANNOT MOVE FURTHER UPSTREAM AND PRESSURE AT THERE CONTINOUSLY INCREASES IN THAT REGION AND IN THIS FORM , A HIGH PRESSURE SHOCK WAVE RAPIDLY FORMS.

THEY ARE NOT LIKE CONVENTIONAL SOUND WAVES , HERE THEY TAKE A FORM OF VERY SHARP CHANGE IN THE PROPERTIES OF GAS . PHYSICALLY , THEY ARE HEARD AS LOUD \"CRACK \" OR \"SNAP\" VOICE  IN AIR . WHEN THIS SHOCK WAVE PASSES THROUGH A LONG DISTANCES , IT CHANGES FROM NON-LINEAR WAVE TO A LINEAR WAVE BY COMING BACK TO IT'S CONVENTIONAL SOUND WAVE BY LOOSING ENERGY IN THE FORM OF HEATING THE AIR . THE SOUND WAVE IS HEARD AS "THUD" OR "THUMP" OF SONIC BOOM WHICH IS CREATED BY SUPERSONIC FLIGHT OF AIRCRAFT .

 

                                                                                                                    

                                                                                                           

 

 

EFFECT OF SGS ON SHOCK LOCATION(SUBSONIC CASE) :-

THE SGS HELPS IN GETTING MORE ACCURATE RESULT OF PHYSICAL QUANTITY . IN SUPERSONIC CASE , AS FLUID FLOWS WITH HIGHER MACH NUMBER , IT GETS EXPANDS IN THE FORM OF EXPANSION WAVE (EXPANSION FAN) DUE TO SUDDEN SHOCK AT THE EDGE OF CHANGE IN AREA . THIS EXPANSION WAVE PRODUCES A WIDE RANGE IN VALUE OF PHYSICAL QUANTITY . SO IN ORDER TO CAPTURE ALL VARIED PHYSICAL QUANTITY VALUE , SGS PLAYS A SIGNIFICANT ROLE. THE SGS WHAT DOES , IT DIVIDES THE WHOLE GRID SCALE INTO SCALE OF SEMI-GRID OF MULTIPLE LEVELS LAYER . THIS SEMI-GRID GENERATES ACCORDING TO ADAPTIVE MESH REFINEMENT WHICH ARISES IN THAT REGION WHERE PHYSICAL QUANITY CHANGES A LOT . IN THIS WAY WE GET A WIDE RANGE OF VALUES IN SMALL SCALE IN EXPANSION WAVE REGION .

 

EFFECT OF SEMI GRID SCALE (SGS) ON CELL COUNT :-

THIS CAN BE UNDERSTAND BY THE FOLLOWING GIVEN BELOW PLOTS :-

SUPERSONIC CASE :-

FOR SGS=0.1 m

 

                                                    

                                                              

 

FOR SGS=0.05 m

 

                                                 

                                                                 

 

FOR SGS=0.02 m

 

                                               

 

                                                                

SUBSONIC CASE :-

FOR SGS=0.1 m

 

                                              

                                                                 

 

FOR SGS=0.05 m

 

                                            

                                                              

 

FOR SGS=0.02 m

 

                                            

                                                              

   

HERE IN ALL THREE CASES OF SUPERSONIC CASE , WE OBSERVE THAT CELL COUNT GOES ON INCREMENT AS WE DECREASE THE SGS PARAMETER . THIS IS BEACUSE FINER MESH LEADS TO GENERATION OF MORE NUMBER OF CELLS BUT SINCE THE MESH IS REFINING WITH ADAPTION TO CHANGE IN VARYING PHYSICAL QUANTITY THEREFORE WE WOULD OBSERVE THAT CELL COUNT IS VARYING IN ZIGZAG FORM (NOT LINEARLY) . 

HERE IN ALL THREE CASES OF SUBSONIC CASE , WE OBSERVE THAT CELL COUNT ALSO INCREASES IN THIS CASE BUT IN A SPECIFIC MANNER . THIS IS BECAUSE WE HAVE PROVIDED DIFFERENT PREDEFINED CELLS IN AMR FOR DIFFERENT SGS PARAMETER THAT\'S WHY WE ARE VIEWING DIFFERENT VARYING PLOTS OTHERWISE IF ALL SGS HAD SAME CELLS THEN DEFINETLY WE WOULD OBSERVED NO VARIATION OR A SMALL VARIATION IN PLOT .

 

CONCLUSION :-

HERE WE UNDERSTAND THAT WHETHER IT IS SUBSONIC OR SUPERSONIC CASE , THE TEMPERATURE WILL RISE  WHEN FLOW MOVES WITH HIGHER VELOCITY . SO THERE IS NO EFFECT OF CASES ON TEMPERATURE AND IT WILL DIRECTLY INCREASE AS FLOW MOVE WITH HIGHER VELOCITY .  THIS CAN BE PROVE BY THIS RELATION ,

                                                                                                                                             `T_(o)/T=(1+(gamma-1)/2Ma^2).`

ALSO , WE UNDERSTAND THAT VELOCITY INCREASES AND PRESSURE DECREASES AS AREA DECRESES FOR SUBSONIC CASE WHEREAS VELOCITY DECREASES AND PRESSURE INCREASES AS AREA DECRESES FOR SUPERSONIC CASE . THIS CAN BE SATISFY BY THE RELATION ,

                                                `(dA)/A=-(dv)/v(1-Ma^2)`

HERE THE RELATION DEPENDS UPON THE CONDITION OF MACH NUMBER THAT WHETHER IT IS IN SUBSONIC OR SUPERSONIC CASE .

HERE WE ALSO CONCLUDED THAT AS SGS PARAMETER WILL DECREASE , IT WILL LEAD TO GENERARTION OF MORE CELLS WHICH WILL BE REFLECT BACK IN CELL PLOT . AND ALSO THE ADAPTIVE MESH REFINEMENT WILL LEAD TO SUDDEN CHANGE IN CELL COUNT .

 

APPLICATION OF PRANDTL MEYER SHOCK :-

IT'S APPLICATION IS WIDELY OBSERVED IN AERODYNAMICS WORLD WHERE WE OBSERVE THAT WHEN AN AIRPLANE ATTAINS SUPERSONIC SPEED THEN THERE IT PRODUCES A SONIC BOOM SOUND AND LEAVES SPEED OF AIR BEHIND IT .