<style> .reveal { font-size: 22px; } </style> ## CFD crash course ### Recognizing the bias in flow modeling <br/> ### --- Session ii - Handling the Flow Movement --- --- ## Similar Flow types, different handling <br/> Content - Coupling the __Continuity__ and __Momentum__ equations - Incopressible and Compressible flow type -- the physics and governing transport - Between compressible and incompressible regime -- the Boussinesq approximation - The __Heat Transfer__ category - Flow over heated/cooled body - Solid body cooled/heated by the flow contribution - CHT: Conjugated Heat Transfer --- #### --- Coupling the __Continuity__ and __Momentum__ equations --- ### Solving flow movement for incompressible flow <br/> Continuity and Momentum conservation in incompressible regime: ($\rho = \text{const}$) $$ \begin{align} \frac{\partial u_j}{\partial x_j} & = 0\ \\ \frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j} u_i u_j & = -\frac{1}{\rho}\frac{\partial P}{\partial x_i} + \frac{\partial}{\partial x_j}\left( \nu \frac{\partial u_i}{\partial x_j}\right) \end{align} $$ <div style="text-align: left "> <br/> &emsp;&emsp;What are the two equations solving? &emsp;&emsp;Is there any conflict between the two equations? </div> ---- #### --- Coupling the __Continuity__ and __Momentum__ equations --- ### Between Pressure and velocity <div style="text-align: left "> General thoughts: 1. Solve the momentum equation for a __provisional velocity__ based on the velocity and pressure of the previous outer loop. 2. Plug the new newly obtained velocity into the continuity equation to obtain a correction. <br/> Tools we need: 1. Pressure-correction method (pressure equation) - conbine the continuity equation to the provisional velocity based pressure representation - solve the pressure equation to acquire the update value for pressure. (Continuity is automatically fulfilled)e.g. [Rhie-Chow interpolation](https://www.cfd-online.com/Wiki/Rhie-Chow_interpolation) 2. Process recipe obtaining the equation solving procedure - Generally for steady state solution: - [SIMPLE -- semi-implicit for pressure linked equation](https://www.cfd-online.com/Wiki/SIMPLE_algorithm) - [SIMPLEC -- semi-implicit for pressure linked equation consistent](https://en.wikipedia.org/wiki/SIMPLEC_algorithm) - Unsteady state relevant - [PISO -- pressure implicit of splitted operators](https://en.wikipedia.org/wiki/PISO_algorithm) - [PIMPLE -- PISO-SIMPLE coupling](https://doc.cfd.direct/notes/cfd-general-principles/the-pimple-algorithm) </div> ---- #### --- Coupling the __Continuity__ and __Momentum__ equations --- ### SIMPLE loops -- steady state incompressible flows <img src="https://www.researchgate.net/publication/306447438/figure/fig3/AS:398690322796548@1472066588738/Overall-stages-in-the-SIMPLE-algorithm-adapted-with-permission-from-Versteeg-and.png" width="50%"> ---- #### --- Coupling the __Continuity__ and __Momentum__ equations --- ### PISO loops -- unsteady state incompressible flows <img src="https://www.researchgate.net/publication/228526451/figure/fig2/AS:301914710396928@1448993484070/PISO-algorithm-flow-chart.png" width="40%"> ### &emsp; &emsp; &emsp;&emsp;Next time step ---- #### --- Coupling the __Continuity__ and __Momentum__ equations --- ### SIMPLE v.s. PISO loops in incompressible flow - It's not forced to utilize SIMPLE for steady state. The algorithm can also be applied in unsteady simulation. The calculational cost is comparatively high. - SIMPLEC, SIMPLER are the improved algorithms based on the same concept. - In PISO algorithm, no relaxation damping is allowed for unsteady state. - Relxation requires loops within one time step &rarr; not available for PISO - The time derivative term $\large \left(\frac{\partial u_i }{\partial t}\right)$ could stablize the numerical instability. - Applying SIMPLE loop within one PISO timestep commonly utilized &rarr; PIMPLE loop <br/> <br/> <br/> <br/> <div style="text-align: left "> &emsp;&emsp;&emsp;How is it for compressible flows? </div> ---- #### --- Coupling the __Continuity__ and __Momentum__ equations --- ### Solving Velocity for __compressible flow__ Specific pressure correcture not necessary, since the $\rho$ is involed in continuity equation. &emsp;&emsp;&rarr; density based solver <br/> <div style="text-align: left "> &emsp;Both methods are applicable for compressible flows | Pressure based solver | Density based solver | | :--------- | :------------ | | 1. Solve __momentum equation__ of intermediate velocity <br/> 2. Calculate intermediate density via __equation of state__ <br/> 3. Apply pressure equation (Rhie-Chow) to acquire correction value in relation to __continuity equation__ <br/> 4. Update pressure, velocity and density <br/> 5. Solve __energy equation__ for temperature T <br/> 6. Loop to converge <br/> <br/>&rarr; SIMPLE/PSIO extension to compressible flow regime | 1. Solve equations for __continuity__ ($\rho$), __momentum__ ($u_i$), __energy__($e$) simultaneously. <br/> 2. Apply __equation of state__ for pressure <br/> 3. Loop to converge <br/><br/> &rarr; completely getting rid of the constant density restriction </div> --- #### --- Incompressible and Compressible flow types --- ### the underlying rationale *All the flow can be described as incompressible, but not all the fluids are incompressible.* <br/> | Flow type | Incompressible flow type | Compressible flow type | | ---------| -------- | -------- | | General rules | 1. for Mach < 0.3 <br/> 2. no density change | 1. for Mach > 0.3 <br/> 2. density change involved via heat transfer, gas dynamic effects <br/> (e.g. shock, under/over expansion wave) <br/>or species mixing process | | Equation concerned | __Continuity__ equation, <br/> __Momentum__ equation and <br/> event. __scalar__ transport | __Continuity__ equation, <br/> __Momentum__ equation, <br/> __Energy__ equation, <br/> Equation of __state__ and <br/> event. __species__ transport | | Related Tasks | Laminar and turbulent flows topics, e.g., Aerodynamics| Heat transfer, transonic/sonic flows, combustion etc.| ---- #### --- Incompressible and Compressible flow types --- ### Governing equations for incompressible flows $$ \begin{align} \frac{\partial u_j}{\partial x_j} & = 0\ \\ \frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j} u_i u_j & = -\frac{1}{\rho}\frac{\partial P}{\partial x_i} + \frac{\partial}{\partial x_j}\left( \nu \frac{\partial u_i}{\partial x_j}\right) +g_i \end{align} $$ <br/> - Continuity equation is simplified to flux conservation - Flux flow in = Flux flow out - Pressure term exists only in the gradient form ($\large \frac{\partial P}{\partial x_i}$) - No restriction of equation of state to indicate the absolute value - No phyiscal violation as long as the gradient is correct. - Reference point can be set to 0 [pa]. ---- #### --- Incompressible and Compressible flow types --- ### Boundary condition for incompressible flows <img src="https://i.imgur.com/nrrlWR6.jpg" width= "75%"> | | inlet | outlet | wall | | -------- | -------- | -------- | --------| | $u_i$ | fixed value (__D__) | fixed gradient (__N__) | fixed value (__D__)| | $P$ | fixed gradient = 0 (__N__) | fixed value (__D__) | fixed gradient = 0 (__N__)| || Given velocity, <br/>but no acceleration ($\large\frac{\partial P}{\partial x_i}$) on boundary surface| Situation of the outlet faces is unknown. Velocity will be equal to the cells before and the pressure gradient is to be generated.| Slip/ non-slip condition for velocity. No pressure effects (breaking or accelerating). ---- #### --- Incompressible and Compressible flow types --- ### Governing equations for compressible flows $$ \begin{align} \text{Continuity Eqn:}&\qquad\qquad\qquad \frac{\partial \rho}{\partial t}+\frac{\partial \rho u_j}{\partial x_j} &&= 0\ \\ \text{Momentum Eqn:}&\qquad\qquad\qquad \frac{\partial \rho u_i}{\partial t} + \frac{\partial}{\partial x_j} \rho u_i u_j && = -\frac{\partial P}{\partial x_i} + \frac{\partial}{\partial x_j} \tau_{ij} + \rho g_i \\ \text{Energy Eqn:}&\qquad\qquad\qquad \frac{\partial K}{\partial t} + \frac{\partial}{\partial x_j} K u_j &&=-P\frac{\partial u_i}{\partial x_i} -\tau_{ij}\frac{\partial u_j}{\partial x_i} + \frac{\partial}{\partial x_j}\left(\kappa \frac{\partial h}{\partial x_j} \right)\\ \end{align} $$ <br/> $$ \begin{align} \text{Ideal gas eqn of state:}\qquad\qquad\qquad P &= \rho R T \\ \text{Shear stress:}\qquad\qquad\qquad \tau_{ij} &= \mu\left[\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j}\right)-\frac{2}{3}\frac{\partial u_k}{\partial x_k}\delta_{ij}\right]\\ \text{Total energy:}\qquad\qquad\qquad K &= \frac{1}{2}\rho u_i^2 + \rho h \\ \text{Caloric perfect gas:}\qquad\qquad\quad h &= c_p T \end{align} $$ ---- #### --- Incompressible and Compressible flow types --- ### Internal compressible flows The convergent-divergent nozzle <img src="https://upload.wikimedia.org/wikipedia/commons/1/14/20140608074127%21A_converging-diverging_nozzle_with_reservoir_pressure_fixed.jpg" width="35%" style="float:right"> - Flow driven by pressure difference - Flow type completely depends on the <br/> pressure ratio between $P_0$ and $P_e$ - if pressure ratio exceeds a certain ratio, <br/> flow chokes on throat. i.e. Mach = 1. <br/> Flow can recover to the subsonic state, or <br/>accelerate to supersonice state. - Shock can occur for exit pressure between C and D, <br/> depending on the exit pressure level - Internal supersonic state cannot be achieved <br/> without a __*convergent-divengent nozzle*__ ---- #### --- Incompressible and Compressible flow types --- ### Under and Overexpansion waves <img src="https://www.engapplets.vt.edu/fluids/CDnozzle/fig4.gif" width="35%"> <img src="https://i.ytimg.com/vi/bmLJq5aPBqc/maxresdefault.jpg" width="50%"> - Outer pressure determines if the expansion wave type. - Perfect match of exit pressure to outer pressure mostly hardly possible - Flow types of nozzle exit are completely different. ---- #### --- Incompressible and Compressible flow types --- ### Shock waves <img src="https://upload.wikimedia.org/wikipedia/commons/7/7b/Schlierenfoto_Mach_1-2_Pfeilfl%C3%BCgel_-_NASA.jpg"> - A discontineous interface, in which the pressure is not capable to match together. Normally due to suddenly huge change of flow characteristics. ---- #### --- Incompressible and Compressible flow types --- ### Flow type and boundaries of compressible flows <img src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQIFBxleabyplcVBTzg_86UD7y254YLhY_zEw&usqp=CAU" width="70%"> Thinking of: 1. What kind of fluid are we dealing with here? 2. Which physical properties require boundary conditions? 3. What else gas related characteristics do we need to declare? 4. What does pressure plays a role in the equation system? 5. What would you setup for a De Laval nozzle flow simulation? ---- #### --- Incompressible and Compressible flow types --- ### Scenario I: Rocket engine and De Laval nozzle <img src="https://2.bp.blogspot.com/-fYSvCoZSqDE/WVJW19faDHI/AAAAAAAACD4/jM9QZzL5pjE3F6H2qqGZXJVpYZU9pRdAgCLcBGAs/s1600/mighty-saturn-v-rocket-nozzle-l.jpg" style="transform: scaleX(-1); float:left" width="35%"> <div style="text-align: left "> &emsp;&emsp;Description: &emsp;&emsp;De Laval nozzle is the common device equipped in the rocket propulsion systme. Aiming in extracting high kinetic energy on the nozzle exit for high thrust, choke condition with conical shape is the necessary elements. General procedure for such challenges follows the isentropic condition <br/> <br/> </div> Thinking on simulating such application: 1. How large would be your ideal domain? What is your arguement? 2. What is the flow type? Compressible or incompressible? 3. What are the governing equation? 4. Which physical properties require boundary conditions, and what are the proper setups for them? ---- #### --- Incompressible and Compressible flow types --- ### Scenario II: Gas mixing process in an closed tank <img src="https://i.imgur.com/K7o97lC.png" width="25%" style="float:left"> <div style="text-align: left "> Description: Tank test is the standardized test device for airbag inflators. The protocol of tank test is to deploy a temperated inflator in the closed tank. The tank is set constantly at 23°C. The pressure level in the tank is then measured during the inflator deployment. </div> Thinking on: 1. What is the flow type? Compressible or incompressible? 2. Say the inflated gas is not air. What are the transport equations? 3. How does the mixing process realized in the transport equation? 4. What are the boundary conditions you would set? - __WALL__ for tank and inflator surface is clear - How about __injection surface__? --- #### --- Between the compressible and incompressible regimes --- ### Rayleigh Bernard Convection <img src="https://i.ytimg.com/vi/OM0l2YPVMf8/maxresdefault.jpg" width="50%"> [Video Example to Rayleigh Bernard Convection](https://gfm.aps.org/meetings/dfd-2017/59babd0cb8ac316d38841e58) What is the physics behind? ---- #### --- Between the compressible and incompressible regimes --- ### The temperature driven flow movement <img src="https://pub.mdpi-res.com/symmetry/symmetry-12-00341/article_deploy/html/images/symmetry-12-00341-ag.png?1585774120" width="70%"> 1. Fluid heated/cooled from the lower/upper plane. 2. Hot medium flows goes upwards and cold gas downwards. - heated fluid &rarr; low density; colded fluid &rarr; high density - density difference ($\rho$) incorporated with gravitation ($g$) induces __bouyancy__ 3. Flow movement is driven by temperature induced __bouyancy__. ---- #### --- Between the compressible and incompressible regimes --- ### Boussinesq Approximation for low density variation flow type $$ \begin{align} \frac{\partial u_j}{\partial x_j} &= 0\ \\ \frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j} u_i u_j & = -\frac{1}{\rho_{\text{ref}}}\frac{\partial P}{\partial x_i} + \frac{\partial}{\partial x_j} \tau_{ij} + \beta\left(T_{\text{ref}} - T\right) \ g_i \\ \frac{\partial T}{\partial t} + \frac{\partial}{\partial x_j} T u_j &= \frac{\partial}{\partial x_j}\left(\alpha \frac{\partial T}{\partial x_j} \right)\\ \end{align} $$ - Flow is treated as incompressible flow, i.e. density change negligible - Buoyancy is the source term incorporated with temperature effects - $\beta$ : thermal expansion coefficient - . In liquid form, e.g., - $\beta_{water} = 6.9e^{-5}$ [1/K], - $\beta_{Ethanol} = 2.5e^{-4}$ [1/K] at 20°C - What is the $\beta$ value for in gas form? ---- #### --- Between the compressible and incompressible regimes --- ### Scenario III: cooling tower and the chimeney effects Description: In the power plant cooling system, hot water flows in and becomes spray dropping down to the cold water tank. During this process, the drops cool down or vaporized. Meanwhile, the air is heated. Density drops and the flow goes upwards to drive the convection. <img src="https://qph.cf2.quoracdn.net/main-qimg-13d6c87670ebd0baf45b9b600efc1b39-lq" width="26.5%"> &emsp;&emsp;&emsp;&emsp;<img src="https://www.fansct.com/Resizer/img.aspx?src=/root/content/industrial-cooling-technology/cooling-towers-and-equipment/natural-draft-cooling-towers/mokra-chv-s-pritozenym-tahem_en.jpg&t=14" width="20.5%"> Thinking on: To evaluate the flow speed on the cooling region/ flow rate on the suction region. How would you conduct the simulation? - How would you model the heat source of the vaporization and water cooling? - How would you set the boundary conditions on inlet and outlet? --- #### --- The Heat Transefer Categories --- ### When dealing with heat transfer problems The dimensionless key indices: <div style="text-align: left "> 1. __Nusselt number__ : ratio of __convective__ ($u_jT$ )to __conductive__ ( $\alpha \frac{\partial T}{\partial x_j}|_{\text{surface}}$ ) heat transfer &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;$\large \text{Nu} = \frac{\text{convective heat tranfer}}{\text{conductive heat transfer}}=\frac{h}{\alpha/L}$ &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;h: convective heat transfer coefficient [$\frac{W}{m^2K}$] 2. __Rayleigh number__ : ratio of __thermal diffusion__ ($\frac{\partial}{\partial x_j}(\alpha \frac{\partial T}{\partial x_j})$ ) to __bouyancy__ ($\Delta\rho g_i$) &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;$\large \text{Ra} =\frac{\text{diffusive heat tranfer}}{\text{buoyancy}}=\frac{\rho \beta \Delta TL³g}{\lambda \mu}$ &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;$\mu$: dynamic viscosity [$\frac{kg}{m s}$] 3. __Richardson number__ classifies the ratio between inertial convection and buoyancy: &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;$\large \text{Ri} =\frac{\text{buoyancy}}{\text{Inertial}} = \frac{\beta g L \Delta T}{u_0^2}$ ---- #### --- The Heat Transefer Categories --- ### Natural and Forced convection <img src="https://www.mhhe.com/engcs/mech/cengel/notes/Image187.jpg" width="350"> | Forced Convection | Natural Convection | | -------- | -------- | | Heat transfer driven <br/> by __inertial flow movement__ | Heat transfer dominated <br/> by __buoyancy__ | <br/> | Ri << 1 | O(Ri) ~ 1 | Ri >> 1 | | -------- | :--------: | -------- | | Buoyancy <br/>negligible | Both effects <br/> have to be considered | Heat convection <br/> negligible | ---- #### --- The Heat Transfer Categories --- ### Flow over heated/cooled body Heat transfer from fluid to boundary mainly due to the thermal boundary layer: $\large \dot{q} =\iint \alpha \frac{\partial T}{\partial y}dA$ <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Thermal_Boundary_Layer_Thickness_formed_by_heated_fluid_flow_along_a_plate.jpg/800px-Thermal_Boundary_Layer_Thickness_formed_by_heated_fluid_flow_along_a_plate.jpg" width="60%"> which means: - zero gradient boundary condition for temperature is __no go__!! - flow can be laminar or terbulent - fluid can be in gas, liquid or multiphase form - Buoyancy force might significant, but not necessary, depends on Richardson number ---- #### --- The Heat Transfer Categories --- ### Flow over heated/cooled body Thinking on: 1. zero gradient boundary condition for temperatureis __no go__!! - What are the phyiscal interpretation of Dirichlet and Neumann boundary condition for temperature on wall? - Which type of boundary condition is suitable for the flow over body scenario? 2. flow can be laminar or terbulent - __Wall function__ for temperature equation also necessary. 3. fluid can be in gas, liquid or multiphase form. - What are the corresponding governing equations? What are your criteria? 4. Buoyancy force might significant, but not necessary, depends on Richardson number - What is the buoyancy if the flow is in the __compressible regime__? ---- #### --- The Heat Transfer Categories --- ### Solid body which receives heat from/passes heat to flow <img src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcR-jZF9jl93i6O4EpjsvmfP_xUvQHckJ9LM_4RLKY3M1ALy7aqn9vxBJYMGaCKdCIBMbSA&usqp=CAU" > Thinking on __Solid body heat transfer__: 1. What is the governing equation for heat-up/cool-down of the solid body? 2. The boundary condition for the solid part, shall it be Delichlet or Neumann? - What is your arguement? ---- #### --- The Heat Transfer Categories --- ### Conjugate Heat Transfer: connecting fluid movement and solid body heat transfer <img src="https://i.imgur.com/Z1DricF.png" width="25%"> $\Large=$ <img src="https://i.imgur.com/IPBZAzL.png" width="25%"> $\Large +$ <img src="https://i.imgur.com/AYj6Ngh.png" width="19%"> - Heat transfer conducted by flow over heated/cooled body, incorporated with heat re-distribution in the solid part. - Multiple simulations are parallelly running, categorized by selected region - Governing equation distinguished to fluid from solid and/or additional parts - Information passes from one region to the other as updated boundary condition, vice versa. Thinking on: - What are the boundary conditions on the fluid-solid interface for fluid and solid demain? ---- #### --- The Heat Transfer Categories --- ### Conjugate Heat Transfer: connecting fluid movement and solid body heat transfer <img src="https://i.imgur.com/Z1DricF.png" width="20%"> $\Large=$ <img src="https://i.imgur.com/IPBZAzL.png" width="20%"> $\Large +$ <img src="https://i.imgur.com/AYj6Ngh.png" width="15%"> Thinking on: - What are the boundary conditions on the fluid-solid interface for fluid and solid demain? |interface <br/> boundary conditions| fluid | solid | | -------- | -------- | -------- | | velocity ($u_i$): | slip/non-slip condition | - | | pressure ($P$): | $\frac{\partial P}{\partial x_n}$ = 0 | - | | temperature ($T$): | fixed value | gradient based heat flux | <br/> - __Segragated steps passing temperature / temperature gradient to each other.__ - Fluid region generate flux by __fixed temperature value__ as boundary - Solid takes the flux, calculates __new temperature__ on solid boundary for fluid part. --- #### --- The Heat Transefer Categories --- ### Scnario iv-a: the cooling water for car engines <img src="https://cdn.britannica.com/15/97215-050-451AE022/gasoline-engine-cooling-system.jpg" width="60%" style="display:block; margin:auto;" > <br/> - Cooling achieved by flowing water over hot objects. - water pump determines the flow rate. - Focus on the water temperature in the cooling system. - What are the governing equations for this phyiscal phenomena and application - What are the boundary condition for the water movement? ---- #### --- The Heat Transefer Categories --- ### Scnario iv-b: the cooling of engine <img src="https://engineeringlearn.com/wp-content/uploads/2021/05/Cooling-System-1024x539.jpg" width="70%" style="display:block; margin:auto;" > <br/> - Cooling achieved by moving water over hot objects (engine). - Temperature in combustion chamber ~ 1800K, conducted to the cylinder wall - Focus on the cylinder cooling performance. - How many objects shall be considered? - What are the governing equations for this phyiscal phenomena and application? - What are the boundary condition for the water movement? ---- #### --- The Heat Transefer Categories --- ### Scnario iv-c: the cooling of cooling water <img src="https://www.researchgate.net/profile/Mohammed-Alktranee-2/publication/352038803/figure/fig1/AS:1029971250266114@1622575690157/Schematic-diagram-of-a-conventional-cooling-system-2_W640.jpg" width="55%" style="display:block; margin:auto;" > <br/> - Cooling achieved by moving water over hot objects. - air flow removes the heat of the radiator - Focus on the air flow sucked through the radiator to achieve the aimed water temperature - How many objects shall be considered? - What are the governing equations for this phyiscal phenomena and application? - Waht are the boundary condition for the water movement?