Lab 8: FLUENT: Turbulent Boundary Layer Flow with Convection Objective: The objective of this laboratory is to use FLUENT to solve for the total drag and heat transfer rate for external, turbulent boundary layer flow. In particular, we will investigate the flow of water over a flat plate with constant surface temperature. This flow has been investigated by many researchers [1-5]. Concepts introduced in this lab will include domain requirements for external flows, external flow boundary conditions, and the use of wall functions for modeling turbulent flows and the corresponding grid requirements. Background: For this lab we will consider water flowing at a uniform velocity and temperature over a flat plate with constant surface temperature. The plate has length L = 1.00 m in the direction parallel to the flow and is very wide in the transverse direction. Thus, we will model it as a two-dimensional flow. The water above the plate has uniform velocity U = 10.0 m/s, temperature

€

T∞ = 290 K , and is at a pressure p = 1.00 atm. The surface temperature of the plate is

€

Ts = 310 K . The Reynolds number at the rear of the plate is

€

ReL =ρU L

µ=1.17 ×107 . (1)

where ρ and µ are the density and viscosity of water at the film temperature defined as

€

Tf = T∞ + Ts( ) 2. For this Reynolds number the flow will be turbulent by the end of the plate. Recall that the transition from laminar to turbulent flow typically occurs at a Reynolds number of approximately 5 x 105. We will assume that the leading edge of the plate is rough such that the boundary layer is turbulent over the entire plate.

Figure 1. Schematic of turbulent boundary layer flow over a flat plate at constant temperature.

x

L = 1.00 m

water at U = 10.0 m/s

p = 1.00 atm flat plate at

hydrodynamic and thermal

boundary layers

y

2

The turbulent hydrodynamic boundary layer that forms at the leading edge of the plate grows thicker downstream along the surface of the plate. Recall that a boundary layer is a thin region at the surface where the flow velocity goes from zero (due to the no-slip condition at the surface) to the free stream velocity. The turbulent boundary layer velocity distribution can be approximated by the following power-law profile developed from a curve fit to experimental data

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u(y) =U yδ

⎛

⎝ ⎜ ⎞

⎠ ⎟ 1/ 7

(2)

where y is the vertical coordinate measured upwards from the plate surface and δ is the boundary layer thickness. A correlation for δ for a smooth plate is

€

δx

=0.382Rex

1/ 5 for

€

5 ×105 ≤ Rex ≤108 (3)

where x is the horizontal distance along the plate measured from the leading edge of the plate and the Reynolds number is defined using x as the length scale. For the same Reynolds number range, a correlation for the skin friction coefficient on the plate has been developed

€

Cf x( ) =τw x( )12 ρU

2 =0.0594Rex

1/ 5 for

€

5 ×105 ≤ Rex ≤108 (4)

where τw is the shear stress at the wall and ρ is the fluid density. Note that the skin friction coefficient and the wall shear stress are theoretically infinite at the leading edge where the boundary layer is very thin and then decreases along the plate. Equation (4) can be integrated over the entire plate to calculate the total drag coefficient

€

CD =

1L

τw dxx= 0

L∫12 ρU

2 =0.0743ReL

1/ 5 (5)

which is used to calculate the total drag force

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FD = 12 ρU

2 A CD (6) where A is the total surface area of the plate in contact with the fluid. In addition to the turbulent hydrodynamic boundary layer, a thermal boundary layer also forms at the leading edge of the plate and grows thicker along the surface of the plate. Due to the significant turbulent mixing in the boundary layer region the growth rate of the hydrodynamic and thermal boundary layers will be the same. Thus, Equation (3) can also be used to determine the thickness of the thermal boundary layer. The local Nusselt number for the turbulent boundary layer is given by the following correlation developed by Reynolds et al. [2]

3

€

Nux =hx xk f

= 0.0296 Rex4 / 5 Pr0.6 Ts

T∞

⎛

⎝ ⎜

⎞

⎠ ⎟

−0.4

(7)

where hx is the local convection coefficient, kf is the thermal conductivity of the fluid, and Pr is the Prandtl number. This correlation is valid for

€

5 ×105 ≤ Rex ≤108 and

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0.6 ≤ Pr ≤ 60. Equation (7) can be integrated over the length of the plate to obtain the average Nusselt number

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NuL =h Lk f

= 0.037 Rex4 / 5 Pr0.6 T s

T∞

⎛

⎝ ⎜

⎞

⎠ ⎟

−0.4

(8)

which can be used to calculate the average heat transfer coefficient,

€

h , and the total heat transfer from the plate using

€

q = h A Ts −T∞( ) . (9) Note that all of the above correlations were developed for turbulent flows over a wide range of conditions such as free stream turbulence and actual plate roughness. Thus, they are only an approximation and are typically considered to be accurate to only about ±15% for any particular flow condition. Laboratory: ICEM CFD To create the mesh for our flow field, we will consider what is necessary to model the “real flow” correctly. In particular, because this is an external flow we have to decide how large to make the overall extent of the domain and how to handle the boundary conditions far away from the flat plate. From Equation (3) we can estimate the thickness of the boundary layer at the end of the plate to be approximately 1.5 cm. However, as the boundary layer grows along the plate the fluid has to move upward in the y-direction to satisfy conservation of mass. To allow the boundary layer to grow unconstrained like it would for a real external flow we must make sure there is sufficient area above the plate well beyond the width of the boundary layer to allow for this growth. The best way to test if the extent of the domain is sufficient is to run your simulations on a series of meshes of increasing extent from the solid surfaces until the solution no longer depends on this parameter. For flat plate simulations a domain height equal to the length of the plate will generally be sufficient. Also, the domain should begin before the plate (about 10% of the length of the plate) to correctly simulate the flow at the leading edge. For external simulations of blunt bodies (such as a car) a general recommendation is that the domain be at least 3 body lengths in front of the body and 5 body lengths behind. Also, the body should not be more than about 1.5% of the total cross-sectional area of the domain.

4

0.001

0.01

0.1

1

10

100

0.0 0.2 0.4 0.6 0.8 1.0

x

y

y+ = 300

y+ = 30

y+ = 5

For the boundary condition at the top of the domain, where the velocity should be about the same as the free stream velocity, there are three choices that will work: (1) fixed velocity of U, (2) pressure outlet, or (3) symmetry. Again, if the extent of the domain is sufficient it should not matter which one is used. Generally, the symmetry boundary condition will work the best insuring that the boundary layer is unconstrained for a given domain size. Finally, as with the pipe flow case that we considered in Lab 6 it is desirable to concentrate the mesh in the boundary layer region where the gradients in velocity and temperature are the highest. In fact, due to the strong interaction of the mean flow and turbulence, the numerical results for turbulent flows tend to be more susceptible to grid dependency than those for laminar flows. Thus, we need to carefully control the number of cells in the boundary layer and their location relative to the wall. We do this in terms of a dimensionless variable called the wall unit defined as

€

y + =ρ uτ y

µ (10)

where

€

uτ is the friction velocity defined as

€

uτ =τwρ

(11)

In particular, we need to make sure that the y+ value for the centroid of the cell closest to the flat plate is in the right range for the turbulence model we choose. We will discuss this further when we select turbulence models in the FLUENT section. We can estimate the size of the element along the plate that corresponds to a particular y+ value using Equation (4) for the skin friction coefficient to estimate the friction velocity. The results are shown in Figure 2.

Figure 2. Estimate of y-coordinate versus axial location and y+ value.

x (m)

y (m

m)

y = δ

5

The range from

€

0 < y + < 5 is called the viscous sublayer because the effects of molecular viscosity dominate in this region. The velocity in this region is simply

€

u+ =uuτ

= y + (12)

The range from

€

5 < y + < 30 is called the buffer layer. There is no simple model to describe the velocity in this range. Next is the log-law layer (or logarithmic overlap layer) where shear stresses due to both molecular viscosity and turbulent mixing are both important. This layer extends from

€

y + > 30 up to approximately

€

y + ≈ 300 where the actual

€

y + magnitude is dependent on the details of the flow. The velocity distribution in this region is given by

€

u+ =uuτ

= 2.5 ln y +( ) + 5.0 (13)

Finally, at higher

€

y + values is the outer turbulent layer where turbulent shear dominates. To begin we will create a mesh where

€

30 < y + < 300 for the centroid of the wall adjacent cell which is in the log-law layer. Using Figure (2) as a guide, we will select a mesh size of 0.2 mm for the cell adjacent to the wall for our initial mesh. After we have completed our simulations we will calculate our observed y+ values along the flat plate to determine if they are in the estimated range. Also, we will insure that we have at least a few cells in the boundary layer region and that we do not use excessive stretching in the direction normal to the wall. We are now ready to create a mesh for the external flow over the flat plate by following the sequence of commands listed below. Your mesh will be a two-dimensional slice of the domain as shown in the Figure 3.

Figure 3. Schematic of computational domain.

L = 1.0 m

H = 1.0 m

y

pressure outlet

velocity inlet

flat plate

L0 = 0.1 m

x

symmetry plane

symmetry plane

6

To run ICEM CFD, click on the ICEM CFD icon on the desktop. In the Main Menu, from the Settings pull down menu select Product Solver. In the DEZ verify under Product Setup Output Solver that ANSYS Fluent Solvers - CFD Version is selected. If it is not, do so, click OK, exit the program, and then restart ICEM CFD. Step 1. Select Working Directory and Create New Project Main Menu - Create a folder for your project. Do not use a name with spaces, including all the directories in the path. From File pull down menu, select Change Working Directory using LMB. In Browse for Folder dialog box select the folder you just created and click OK. Verify that the new working directory has been set in the Message Window. Main Menu - From File pull down menu, select New Project using LMB. In New Project dialog box create a new project. Again, do not use a name with spaces. Verify that your new project has been created in the Message Window. Step 2. Start Recording Replay Script Because we will create several meshes, we will use a replay script (written in the scripting language Tcl/Tk) to generate meshes for a range of domain heights automatically. Main Menu - From File pull down menu, select Reply Scripts -> Replay Control using LMB. Organize the ICEM CFD window and Replay Control window so you can see both and ensure that Record (after current) is selected in the Replay Control window. While creating the geometry and mesh in the steps below note that script lines (or instructions) will be automatically recorded in the Replay Control window under Operations in script. Step 3. Create Points for Geometry Function Tab - From Geometry select Create Point using LMB. DEZ - For Create Point enter the following: deselect Inherit Part (NOTE, this is only needed for Windows OS), in Part text edit box click LMB and enter PNT (replacing GEOM),

select Explicit Coordinates using LMB, under Explicit Locations ensure Create 1 point is selected from pull down menu, in X, Y, and Z text edit boxes ensure 0 is entered in each for point at (0, 0, 0), click Apply using LMB and verify the Message Done: points pnt.00, repeat this process for all of the points listed in Table 1, and click Dismiss using LMB.

Utilities - Select Fit Window using LMB to verify that nine points have been created. DCT - Expand Geometry and Parts menus by using LMB to change + to - for each.

7

Under Model\Geometry use RMB to click on Points and select Show Point Names using LMB. Verify that nine points have been created.

Table 1. Coordinates for points with part name PNT. Name x (mm) y (mm) z (mm) pnt.00 0 0 0 pnt.01 -100 0 0 pnt.02 1000 0 0 pnt.03 1000 50 0 pnt.04 0 50 0 pnt.05 -100 50 0 pnt.06 -100 1000 0 pnt.07 0 1000 0 pnt.08 1000 1000 0

In the Replay Control Window click Clean and Renumber using LMB. Note that lines that mark the beginning and ending of an undo group are deleted. Step 4. Create Curves for Geometry

Function Tab - From Geometry select Create/Modify Curve using LMB. DEZ - For Create/Modify Curve enter the following: deselect Inherit Part (NOTE, this is only needed for Windows OS), in Part text edit box click LMB and enter WALL (replacing PNT),

select From Points using LMB, select Select location(s) using LMB (if not already selected), select pnt.00 and pnt.02 using LMB and then click MMB to create flat plate, verify the Message Done: curves crv.00, in Part text edit box click LMB and enter INLET, select pnt.01 and pnt.05 using LMB and then click MMB to create lower inlet, verify the Message Done: curves crv.01, select pnt.05 and pnt.06 using LMB and then click MMB to create upper inlet, verify the Message Done: curves crv.02, in Part text edit box click LMB and enter OUTLET, select pnt.02 and pnt.03 using LMB and then click MMB to create lower outlet, verify the Message Done: curves crv.03, select pnt.03 and pnt.08 using LMB and then click MMB to create upper outlet, verify the Message Done: curves crv.04,

8

in Part text edit box click LMB and enter SYMMETRY, select pnt.01 and pnt.00 using LMB and then click MMB to create boundary before plate, verify the Message Done: curves crv.05, select pnt.06 and pnt.07 using LMB and then click MMB to create left upper boundary, verify the Message Done: curves crv.06, select pnt.07 and pnt.08 using LMB and then click MMB to create right upper boundary, verify the Message Done: curves crv.07, in Part text edit box click LMB and enter INTERIOR, select pnt.04 and pnt.00 using LMB and then click MMB, verify the Message Done: curves crv.08, select pnt.04 and pnt.07 using LMB and then click MMB, verify the Message Done: curves crv.09, select pnt.04 and pnt.05 using LMB and then click MMB, verify the Message Done: curves crv.10, select pnt.04 and pnt.03 using LMB and then click MMB, verify the Message Done: curves crv.11, click DISMISS using LMB. DCT - Under Model\Geometry use RMB to click on Points and unselect Show Point Names using LMB. Under Model\Geometry use RMB to click on Curves and select Show Curve Names using LMB. Verify that twelve curves have been created. In the Replay Control Window click Clean and Renumber using LMB. Note that lines that mark the beginning and ending of an undo group are deleted. Step 5. Create Surfaces for Fluid Flow Function Tab - From Geometry select Create/Modify Surface using LMB. DEZ - For Create/Modify Surface enter the following: ensure Inherit Part is NOT selected, in Part text edit box click LMB and enter FLUID (replacing INTERIOR),

select Simple Surface using LMB, under Surf Simple Method ensure From 2-4 Curves is selected from pull down menu, select Select curve(s) using LMB, select crv.00, crv.03, crv.08, and crv.11 using LMB and then click MMB, select crv.01, crv.05, crv.08, and crv.10 using LMB and then click MMB, select crv.02, crv.06, crv.09, and crv.10 using LMB and then click MMB, select crv.04, crv.07, crv.09, and crv.11 using LMB and then click MMB, and click DISMISS using LMB. NOTE: This step is not necessary for creating our mesh because the required surfaces are also created during the blocking step below.

9

Step 6. Create Blocking To create the meshes in Labs 6 and 7 we used 2-D surface (or shell) meshing. For this lab to create the mesh we will use blocking, a useful tool for creating 2-D and 3-D structured grids for complicated geometries. Step 1 is to create a block (or rectangle for 2-D) around the entire geometry, consisting of edges and vertexes, which can be split up into sub-blocks (some of which can be deleted) depending on the requirements of the geometry. Step 2 associates the block vertexes and edges with the actual geometry points and curves, respectively. Step 3 specifies the distribution of nodes along each edge from which a structured mesh is generated for each block and then mapped to the actual geometry. Figure 4 shows an example for a simple pipe flow. For step 1, an initial single block is split twice vertically and once horizontally. Then, the bottom left and right blocks are deleted leaving four sub-blocks as shown on the left. For step 2, the vertexes and edges of the block are associated with the actual pipe geometry on the right such that the bottom sub-block corresponds to the small pipe and the three top blocks corresponded to the large pipe. For step 3, node distributions on each edge of the sub-blocks are specified, structured meshes are created on each sub-block, and then mapped to the actual pipe flow as shown.

Figure 4. Example of a block mesh mapped onto a curved geometry.

For this lab, the blocking is very simple and our block and actual geometry overlap as shown in Figure 5 allowing us to use auto association. Blocking is still useful for this lab because it allows us to easily modify the domain extent without changing the boundary layer mesh.

!

10

Figure 5. Schematic of blocking strategy for flat plate. DCT - Under Model\Geometry use RMB to click on Curves and unselect Show Curve Names using LMB, use RMB to click on Surfaces and select Show Surface Names using LMB and verify that four surfaces have been created, and then unselect Curves and Surface to hide them.

Function Tab - From Blocking select Create Block using LMB. DEZ - For Create Block enter the following: under Part select FLUID from pull down menu, under Initialize Blocks Type select 2D Planar from pull down menu, and click Apply and Dismiss using LMB.

Function Tab - From Blocking select Split Block using LMB. DEZ - For Split Block enter the following: under Split Method select Prescribed point from pull down menu, select Select edge(s) using LMB, select left edge using LMB, select point on left edge at ( -100, 50, 0) using LMB to create horizontal split, select bottom edge using LMB, select point on bottom edge at (0, 0, 0) using LMB to create vertical split, click Dismiss using LMB. NOTE: For the block, boundary edges are colored black and interior edges are light blue. Function Tab - From Blocking select Associate using LMB.

Blocking

Figure: Blocking Menu

The Blocking tab contains the following options to create blocking over any geometry.Create BlockSplit BlockMerge VerticesEdit BlockAssociateMove VertexTransform BlocksEdit EdgePre-Mesh ParamsPre-Mesh QualityPre-Mesh SmoothBlock ChecksDelete Block

Create Block

Figure: Create Block Options

The following options are available for creating blocks:Initialize BlockFrom Vertices/FacesExtrude Face2D to 3D Blocks3D to 2D

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Figure: 3D Blocking

Figure: 2D Blocking

Split Block

The following options are available for splitting blocks:Split BlockOgrid BlockExtend SplitSplit FaceSplit VerticesSplit Free FaceImprint Free Face

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Blocking

Associate

Figure: Blocking Associations Options

The following options are available for blocking associations.Associate VertexAssociate Edge to CurveAssociate Edge to SurfaceAssociate Face to SurfaceDisassociate from GeometryUpdate AssociationsReset AssociationsSnap Project VerticesGroup/Ungroup CurvesAuto Associate

Associate Vertex

The Associate Vertex option allows you to associate vertices and project the vertex onto itself, points,curves, and surfaces. Select the vertex and the entity to project it onto.

Associate Edge to Curve

The Associate Edge to Curve option allows you to associate the edges of blocks to curves. The vertices atthe end of the edges are also associated to the same curve unless they were previously associated to anothercurve or point. Edge segments can be individually associated after using edge splits. Multiple edges can beassociated with multiple curves, but all the curves will be grouped into a single composite curve.

Note

Associating edges to curves also results in the creation of line elements along those curves. For2D planar blocking, it is essential that all the perimeter edges be associated with perimeter curvesbecause many solvers use the perimeter line elements as boundaries.

Project verticesIf enabled, the vertices will automatically be projected to the corresponding curves.

Project to surface intersectionIf enabled, the surface-surface intersection will be captured correctly. This is for poor geometry situationswhere the intersection curve may not match with the intersection of the surfaces. If you associate anedge with a curve using this option, the edge will be colored purple. The edge is associated with the

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formation of ANSYS, Inc. and its subsidiaries and affiliates.

Associate Edge to Curve

L

H

y

L0

x pnt.02 pnt.00 pnt.01

pnt.05 pnt.04 pnt.03

pnt.07 pnt.08 pnt.06

h crv.00

crv.11

crv.07

crv.05

crv.10

crv.06

crv.08

crv.09

crv.01

crv.02 crv.04

crv.03

11

DEZ - For Blocking Associations enter the following:

under Edit Associations select Auto Association using LMB, under Auto Association ensure Snap Project Vertices is selected, and click Apply and Dismiss using LMB. NOTE: All the block edge colors should turn to green to indicate they have been associated with a curve. Using auto association is only possible because of the very simple geometry. In Lab 9 we will need to manually associate each vertex and edge. Step 7. Mesh Blocks and Surface

Function Tab - From Blocking select Pre-Mesh Params using LMB. DEZ - For Pre-Mesh Params enter the following:

under Meshing Parameters select Edge Params using LMB, scroll down and select Copy Parameters using LMB under Copy Method select To All Parallel Edges from pull down menu, scroll up and select Select Edges(s) using LMB, select bottom edge on left (corresponding to before the flat plate) using LMB, under Mesh law select Uniform from pull down menu, under Nodes enter 11, NOTE: The edge meshing will automatically be applied, recorded in the replay script, and shown as soon as you enter or change the number of nodes so you do not need to click Apply. select Select Edges(s) using LMB, select bottom edge on right (corresponding to flat plate) using LMB, under Mesh law select Uniform from pull down menu, under Nodes enter 101 select Select Edges(s) using LMB, select left edge on bottom (corresponding to flow inlet on bottom) using LMB, under Mesh law select Geometric 1 from pull down menu, under Spacing 1 enter 0.2 for the spacing for the first nodes from the surface, under Nodes enter 22 (for Ratio 1 of 1.20816 and Spacing 2 of 8.78027 as indicated), select Select Edges(s) using LMB, select left edge on top (corresponding to flow inlet on top) using LMB, under Mesh law select Uniform from pull down menu, under Nodes enter 96 (corresponding to an element size of 10 mm), click Dismiss using LMB. DCT - Expand Blocking menu by using LMB to change + to - . Under Model\Blocking select Pre-Mesh. In the Mesh Dialog Box select Yes to compute the mesh.

Selectedgroups only the selected curves.

All tangentialgroups all curves that are tangential to the selected curves.

Part by Partgroups all the curves in each part into composite curves.

Ungroup Curvesallows you to select the composite curve to ungroup.

Auto Associate

The Auto Associate option allows you to associate edges to curves. Auto Association looks at the topologyof the surfaces and the topology of the blocking and attempts to link the edge projections of the blockingin relation to the topology of the geometry.

Move Vertex

Figure: Move Vertices Options

The following options are available for moving vertices:Move VertexSet LocationAlign VerticesAlign Vertices In-lineSet Edge LengthMove Face Vertices

Move Vertex

The Move Vertex option allows you to modify the location of a vertex. Select a vertex using the left mousebutton, accept the selection by pressing the middle mouse button, and use the right mouse button tocancel the selection. After fixing the constraints, select the vertex to move.

Single Methodallows you to select a single vertex to be moved.

Multiple Methodallows you to select multiple vertices for movement.

Movement Constraintsallows you to constrain a vertex to moving in any direction.

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Move Vertex

Unsplit Edge

The Unsplit Edge option allows you to remove splits from edges. For the Single method, select the vertexof the split edge to remove. For the All method, select the edge to remove all the splits.

Link Edge

The Link Edge option allows you to set the shape of edges. Select the target edges and then the edge forthe source of the shape.

Selectedsets the shape of only the selected edge.

In dimensionsets the shape automatically for all connected edges. Select the source edge for the Link Edge Dimension,the Source index or vertex, and the Target index or vertex.

Unlink Edge

The Unlink Edge option allows you to unset the shapes of edges linked by the Link Edge option.

Pre-Mesh Params

Figure: Pre-Mesh Parameters Options

The following options are available for Pre-Mesh parameters:Update SizesScale SizesEdge ParamsMatch EdgesRefinement

Update Sizes

The Update Sizes option allows you to update sizes in the pre-mesh. The following options are availablefor updating sizes in the pre-mesh:

Update Allcomputes the edge node spacing based on constraint equations with the default (BiGeometric) meshinglaw. You can adjust the number of nodes on each edge based on the Global Surface or Curve MeshSize and by default each edge will follow the BiGeometric geometry law.

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Blocking

Scale Sizes

The Scale Sizes option allows you to specify the factor by which the mesh size should be globally multiplied.After clicking Apply, you will be asked whether to recompute the mesh.

In Figure: Block Mesh Scaled by Factor of Two (p. 377), the mesh size (from Figure: Initial Block Mesh (p. 377))was changed by a factor of 2.

Figure: Initial Block Mesh

Figure: Block Mesh Scaled by Factor of Two

Edge Params

The Edge Params option allows you to modify the mesh parameters in a detailed manner by specifyingvarious bunching laws and the node spacing along any particular edge. Each edge has several parametersthat determine the spacing of the mesh along the edge: number of nodes, the meshing law, initial lengthat the beginning/end of the edge, the expansion of the mesh from the beginning/end of the edge to theinterior, and the maximum element length along the edge.

The Edge Params icon brings up a window with all the mesh parameters. Once an edge has been selected,the mesh parameters for that edge will be displayed. All parameter values may be modified, except for theEdge ID and the Edge Length, which are pre-defined.

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Edge Params

12

NOTE: You should produce a structured mesh with nodes concentrated near the wall. Step 8. Save Files and Export Initial Mesh In Replay Control window unselect Record (after current) to stop recording your script. Click Save and use the Save Script File Dialog Box to save a copy of your script file. Main Menu - From File pull down menu, select Blocking -> Save Unstructured Mesh using LMB. Use the Save Mesh as Dialog Box to save the unstructured mesh. Function Tab - From Output Mesh select Output To Fluent V6 Boundary Cond. using LMB. In Family Part boundary conditions dialog box: expand Edges and Mixed/unknown menu by using LMB to change + to -, expand INLET menu by using LMB to change + to -, click Create new to open the Selection dialog box, under Boundary Conditions select velocity-inlet using the LMB, click Okay using LMB to close the Selection dialog box, expand INTERIOR menu by using LMB to change + to -, click Create new to open the Selection dialog box, under Boundary Conditions select interior using the LMB, click Okay using LMB to close the Selection dialog box, expand OUTLET menu by using LMB to change + to -, click Create new to open the Selection dialog box, under Boundary Conditions select pressure-outlet using the LMB, click Okay using LMB to close the Selection dialog box, expand SYMMETRY menu by using LMB to change + to -, click Create new to open the Selection dialog box, under Boundary Conditions select symmetry using the LMB, click Okay using LMB to close the Selection dialog box, expand WALL menu by using LMB to change + to -, click Create new to open the Selection dialog box, under Boundary Conditions select wall using the LMB, click Okay using LMB to close the Selection dialog box, click Accept using LMB. Function Tab - From Output Mesh select Write Input using LMB. In Save dialog box click Yes using LMB to Save current project first. In Open dialog box click Open to select unstructured mesh with current project name. In ANSYS Fluent V6 dialog box enter the following: in Grid dimension select 2D using LMB, in Scaling ensure No is selected,

13

in Write binary file ensure No is selected, in Ignore couplings ensure No is selected, in Boco file retain the default file name, in Output file change the file from fluent to a new name for your first mesh, and click Done using LMB. Step 9. Create and Save Meshes with Different Boundary Layer Node Spacings Return to Steps 7 and 8, modify the edge parameters using the settings for Cases 2 through 5 shown in Table 2 (you have already done Case 1), and save your new project and meshes to separate files. To recreate the mesh each time in the DCT under Model and Blocking deselect and then reselect Pre-Mesh.

Table 2. Mesh settings for different boundary layer element spacing. All meshes have a total height of 1.0 m.

Case Spacing 1 (mm) Nodes Growth

Factor Spacing 2

(mm) 1 0.2 22 1.20816 8.78027 2 0.002 40 1.25139 10.0462 3 0.2 35 1.10082 4.75106 4 0.2 17 1.31472 12.1213 5 0.2 14 1.43545 15.3071

Step 10. Create and Save Meshes with Different Domain Heights You will create 3 new meshes with new domain heights as shown in Table 2 where the bottom two blocks are kept the same (Case 1 in Table 1) and only change the top two blocks are changed. You will automate this process by using an edited replay script. You could use your own replay script, but instead use the one named lab_8.rpl available from my web page and in Appendix A. It is similar to the one you created, except that all unnecessary lines have been removed, comment lines have been added, geometry parameters are defined to make it easier to change the height, and lines to set the boundary conditions are included at the end.

Table 2. Mesh settings for different heights.

Case Height (mm) Nodes

6 500 46 7 250 21 8 100 6

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Complete the following steps to test the original replay script and then to change the height and number of nodes to the values shown in Table 2 to generate your new meshes: Main Menu - From File pull down menu, select Close Project using LMB. Use the Save Dialog Box to save the current project if necessary. In Replay Control window click Load using LMB to open text editor. Under Operations in Script highlight the first line and click Do all using LMB. In DCT under Model expand Blocking and select Pre-Mesh using LMB to display mesh. Verify that your Case 1 mesh has been recreated. Main Menu - From File pull down menu, select Close Project using LMB. Use the Save Dialog Box to not save the current project (you already have your own). In Replay Control window click Edit using LMB to open text editor. Note that lines 5 through 10 are used to set parameter values for the geometry. Change the height, H, to 500 for Case 6 above. Notice that in line 65 of the replay script, the number of nodes necessary to keep the mesh spacing the same (dx = 10 mm) is calculated. Save and exit the replay script. Under Operations in Script highlight the first line and click Do all using LMB. In DCT under Model expand Blocking and select Pre-Mesh using LMB to display mesh. A new mesh with the new height but the same element sizes on the upper portion should be created. Main Menu - From File pull down menu, select Blocking and Save Unstructured Mesh using LMB. Use the Save Mesh As Dialog Box to save the current unstructured mesh. Main Menu - From File pull down menu, select Save Project using LMB. Use the Save Project Dialog Box to save the current project. Function Tab - From Output Mesh select Output To Fluent V6 Write Input using LMB. In Save dialog box click Yes using LMB to Save current project first. In Open dialog box click Open to select unstructured mesh with current project name. In ANSYS Fluent V6 dialog box enter the following: in Grid dimension select 2D using LMB, in Scaling ensure No is selected, in Write binary file ensure No is selected, in Ignore couplings ensure No is selected, in Boco file retain the default file name, in Output file change the file from fluent to a new name for your first mesh, and click Done using LMB. Repeat these steps for the other 2 heights and node spacings. Main Menu - From File pull down menu, select Exit using LMB.

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FLUENT Similar to the mesh, turbulence models must be carefully selected that are capable of correctly modeling wall interactions so that quantities like wall shear stresses (used to calculate drag) and temperature gradients (used to calculate heat transfer rates) can be accurately predicted. For this class will use the k-ε turbulence model which is an older, but still very popular model for CFD simulations. However, this model in its simplest form is primarily valid for the turbulent core region away from walls. Therefore, the simplest version of the k-ε model must be modified to model the near-wall region. The following are two common approaches used: (1) Wall Function Method: The viscous sublayer and buffer layer are not resolved. Instead, semi-empirical formulas called “wall functions” (Equations (12) and (13) for the velocity) are used to bridge the viscosity-affected regions between the wall and the log-law layer. The use of wall functions eliminates the need to modify the turbulence models to account for the presence of the wall. The wall function method is in general economical, robust, and reasonably accurate. The wall function method is further divided into “standard” and “non-equilibrium” versions. The non-equilibrium version more accurately accounts for the effects of pressure gradients and departures from equilibrium and is recommended for flows with high pressure gradients and separation. The following mesh requirements should be used for this method:

• The wall-adjacent cell’s centroid should be located between

€

30 < y + < 300 which is in the log-law layer. A

€

y + value close to the lower bound (

€

y + ≈ 30) is most desirable.

• Excessive stretching in the direction normal to the wall should be avoided.

• There should be at least about 3-5 cells inside the boundary layer. (2) Near-Wall Modeling: The turbulence models are modified to enable the viscosity-affected region to be resolved with a mesh all the way to the wall, including the viscous sublayer. Because the resulting equations are more complicated and the mesh needs to be more refined near the wall this method is computationally more expensive. However, in situations where the details of the boundary layer need to be resolved or they do not follow the standard relations given by the wall functions this method will be much more accurate. The following mesh requirements should be used for this method:

• The wall-adjacent cell’s centroid should be located between

€

0 < y + < 5 which is in the viscous sublayer. A

€

y + value close to 1 is most desirable.

• Excessive stretching in the direction normal to the wall should be avoided.

• There should be at least 10 cells inside the viscosity-affected near-wall region. Note that for both of these methods it is very important that the wall adjacent cell’s centroid not lie in the buffer region (

€

5 < y + < 30).

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For our case of turbulent flow over a flat plate where there are no strong pressure gradients and no flow separation, any of these models will perform reasonably well. We will demonstrate this by running each of these near-wall modeling approaches and show that we can get reasonable results for each as long as we follow the recommended guidelines for the mesh. Begin by completing the following steps: Step 1. Read In Mesh Import your first mesh created using ICEM CFD into FLUENT. Check to make sure the mesh imported correctly and that you scale it correctly from mm to m. Step 2. Problem Setup for Initial Simulation In the Navigation Pane Tree under Problem Setup use the following steps to setup your simulation: General, Solver • Type: Pressure-Based • Time: Steady • Velocity Formulation: Absolute • 2D Space: Planar Models (remaining models off) • Energy: On • Viscous (use defaults for coefficients and remaining options)

o Model: k-epsilon o k-epsilon Model: Realizable o Near-Wall Treatment: Standard Wall Functions

Materials, Fluid, water (change the properties for water to those at 300 K) • Density: 997 .kg/m3 • Specific Heat: 4179 J/kg•K • Thermal Conductivity: 0.613 W/m•K • Viscosity: 8.55e-04 kg/m•s Cell Zone Conditions • Zone: water-flow

o Type: fluid o Material Name: water

• Operating Conditions o Operating pressure: 101,325 Pa o Gravity: NOT selected

Boundary Conditions • Zone: inlet

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o Type: velocity-inlet o Edit: Momentum tab

• Velocity Specification Method: Magnitude, Normal to Boundary • Velocity Magnitude: 10.0 m/s • Turbulence, Specification Method: Intensity and Viscosity Ratio • Turbulent Intensity: 1% • Turbulent Viscosity Ratio: 1.0

o Edit: Thermal tab • Temperature: 290 K

• Zone: outlet o Type: pressure-outlet o Edit: Momentum tab

• Gage Pressure: 0 Pa, • Turbulence, Specification Method: Intensity and Viscosity Ratio • Turbulent Intensity: 1% • Turbulent Viscosity Ratio: 1.0

o Edit: Thermal tab • Temperature: 300 K

• Zone: wall o Type: wall o Edit: Momentum tab

• Wall Motion: Stationary Wall • Shear Condition: No-Slip

o Edit: Thermal tab • Temperature: 310 K

• Zone: symmetry o Type: symmetry

Step 3: Solution Setup for Simulation In the Navigation Pane Tree under Solution use the following steps to setup your solution methods, controls, monitors, and initialization: Solution Methods • Pressure-Velocity Coupling

o Scheme: SIMPLE • Spatial Discretization

o Gradient: Least Squares Cell Based o Pressure: Standard o Momentum: QUICK o Turbulent Kinetic Energy: QUICK o Turbulent Dissipation Rate: QUICK o Energy: QUICK

Solution Controls • Under-Relaxation Factors: default values

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Monitors, Residuals • Options

o Print to Console: selected o Plot: selected

• Equations, Residual o Monitor: selected for all o Check Convergence: selected for all o Absolute Criteria: 1e-6 for all

Solution Initialization • Compute from: inlet Step 4. Run Calculation Navigation Pane Tree - Under Solution select Run Calculation. Task Page - Under Number of Iterations enter 1000 and click Calculate. The problem should converge after about 500 iterations. However, later cases will not always reach 10-6 residuals. Step 5. Make Contour Plots Visualize your results by displaying a contour plot of temperature. You should see a VERY thin layer of hot fluid near the surface of the plate. You will have to zoom in to see this. Step 6. Make xy Plots of Variables Make an x/y plot of the x-velocity profile at the exit of the flow using the following steps: Navigation Pane Tree - Under Results select Plots. Task Page - Under Plots select XY Plot and Set Up to open Solution XY Plot Dialog Box. In the Solution XY Plot Dialog Box do the following steps: deselect Position on X Axis, select Position on Y Axis, under Plot Direction for X enter 0 and for Y enter 1, under X Axis Function select Velocity and X Velocity, under Surfaces select outlet click Axes button and in the Axes - Solution XY Plot Dialog Box, under Axis select Y, under Options deselect Auto Range, under Range set the Maximum to 0.05, and click Apply and Close click Plot to display the xy plot in the Display Window.

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NOTE: Check that the velocity profile looks like a typical turbulent velocity profile and that the thickness of the boundary layer is about 1.5 cm. Also, check how many nodes appear to be within the boundary layer region and verify that it is at least 3-5. Write the data to a file. Make an x/y plot of the temperature profile at the exit of the flow. To do this follow the instructions above, but under X Axis Function select Temperature and Static Temperature. NOTE: Check that the temperature profile looks like a typical turbulent temperature profile and that the boundary layer thickness is also about 1.5 cm. Write the data to a file. Make x/y plot of dimensionless wall unit, y+, for first cell along plate using the following steps: In the Solution XY Plot Dialog Box do the following steps: deselect Position on Y Axis, select Position on X Axis, under Plot Direction for X enter 1 and for Y enter 0, under Y Axis Function select Turbulence and Wall Yplus, under Surfaces deselect outlet and select wall, click Axes button and in the Axes - Solution XY Plot Dialog Box under Axis select Y, under Options select Auto Range, and click Apply and Close click on Plot. NOTE: Check that y+ values begin at about 65 near the leading edge and then decrease to about 40. This satisfies our mesh requirements for range for this value. Write the data to a file. Step 7. Define Dimensionless Variables First, set the reference values used to calculate the friction coefficient and heat transfer coefficient to the flow conditions far above the plate and the properties set earlier for water using the following steps: Navigation Pane Tree - Under Problem Setup select Reference Values. Task Page - Under Reference Values select the following: under Compute From select inlet and under Reference Zone select fluid from the pull-down menus. Define the local Reynolds number (use “re-x” for the name) and Nusselt number (use “nu-x” for the name) using Define\ Create Custom Field Functions from the Menu Bar User-Defined tab. For both the Reynolds and Nusselt numbers use the x-location for the length scale. To do this under Field Functions select Mesh and X-Coordinate from the pull-down menus.

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For the Reynolds number you can find ρ in the Field Functions under Density and µ in the Field Functions under Properties and Molecular Viscosity where the definition will be given as viscosity-lam. For the velocity use the constant free stream velocity of 10 m/s. For the Nusselt number you can find hx in the Field Functions under Wall Fluxes and Surface Heat Transfer Coefficient where the definition will be given as heat-transfer-coef. Also, you can find kf in the Field Functions under Properties and Thermal Conductivity where the definition will be given as thermal-conductivity-lam. NOTE: FLUENT has a built in function that calculates the Nusselt number, but it uses a constant length scale (defined in the reference values) instead of the distance along the plate which is what we need to compare to correlations. Step 8. Make xy plots of Dimensionless Variables Make an x/y plot of the skin friction coefficient versus Reynolds number. Navigation Pane Tree - Under Results select Plots. Task Page - Under Plots select XY Plot and Set Up to open Solution XY Plot Dialog Box. In the Solution XY Plot Dialog Box do the following steps: deselect Position on X Axis, under Y Axis Function select Wall Fluxes and Skin Friction Coefficient, under X Axis Function select Custom Field Functions and re-x, under Surfaces select wall, and click on Plot. NOTE: The skin friction coefficient should be highest near the leading edge and decrease along the plate. Write the data to a file. Make an x/y plot of Nusselt number versus Reynolds number. In Solution XY Plot Dialog Box do the following steps: deselect Position on X Axis, under Y Axis Function select Custom Field Functions and nu-x, under X Axis Function select Custom Field Functions and re-x, under Surfaces select wall, and click on Plot. NOTE: The Nusselt number should be 0 at the leading edge and increase to approximately 40,000. Write the data to a file. Step 9. Calculate Total Drag Force and Heat Transfer Rate Navigation Pane Tree - Under Results select Reports.

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Task Page - Select Forces and click Set Up to open the Force Reports Dialog Box. In Force Reports Dialog Box under Wall Zones select wall and click Print to display calculated forces in Console. The pressure force should be zero because the plate is parallel to the flow. Record the magnitude of the viscous force. A width of 1.0 m set in the reference values is used for this calculation. Task Page - Select Fluxes and click Set Up to open the Flux Reports Dialog Box. In Force Reports Dialog Box under Options select Total Heat Transfer Rate, under Boundaries select wall, and click Compute to display calculated fluxes in Console. Record the magnitude of the total heat transfer rate. A width of 1.0 m is also used for this calculation. Step 10. Save Case and Data Files Step 11. Test Near-Wall Treatments Use the Viscous Model Dialog Box to change the turbulence model’s Near-Wall Treatment to Non-Equilibrium Wall Functions and rerun the simulation on the current mesh by repeating Steps 4 through 10 (using a different file name) for this case. Use the Define\Model\Viscous menu to change the turbulence model’s Near-Wall Treatment to Enhanced Wall Treatment and rerun the simulation on a revised mesh that has the appropriate refinement in the viscous sublayer (Case 2 in Table 1). Rerun the simulation by repeating Steps 4 through 10 (using a different file name) for this case. In particular verify that the y+ values are again correct for this turbulence model’s wall treatment. Step 12. Test Mesh Refinement Use a series of meshes (Cases 1, 3, 4, and 5 in Table 1) to test the effect of refining the grid in the boundary layer. Rerun the simulation by repeating Steps 4, 9, and 10 (using a different file name) for each case. Make sure to correctly scale the mesh from mm to m each time. Note that since all three near-wall region models perform about the same for this flow you will use the simplest (k-ε turbulence model with standard wall functions) for the comparison. Step 13. Test Domain Height Use a series of meshes (Case 1 in Table 1 and Cases 6, 7, and 8 in Table 2) to test the effect of domain height. Rerun the simulation by repeating Steps 4, 9, and 10 (using a different file name) for each case. Again, make sure to correctly scale the mesh from mm to m each time. Again, since all three near-wall region models perform about the same for this flow you will use the simplest (k-ε turbulence model with standard wall functions) for the comparison.

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Assignment Submit the following figures: Figures 1 (a) and (b). Plots of Case 1 and Case 2 meshes zoomed in to show wall region. Figures 2 (a), (b), and (c). Plots for y+ versus x for the 3 near-wall region turbulence models (Case 1 with Standard Wall Functions, Case 1 with Non-Equilibrium Wall Functions, and Case 2 with Enhanced Wall Treatment). For all 3 near-wall region turbulence models above on the same plot, hand in Figures 3-6:

Figure 3. Plot of x-velocity component versus y at outlet along with power law velocity profile given by Equation (2). Note, Equation (2) is only valid for y = 0 to δ. Figure 4. Plot of temperature versus y at outlet. Figure 5. Plot of skin friction coefficient versus Reynolds number along with correlation for skin friction coefficient given by Equation (4). Figure 6. Plot of Nusselt number versus Reynolds number along with correlation for Nusselt number given by Equation (7).

Figures 7 (a) and (b). Plots of drag force and heat transfer rate versus height of domain, H (corresponding to Cases 1, 6, 7, and 8). Figures 8 (a) and (b). Plots of drag force and heat transfer rate versus growth factor (corresponding to Cases 1, 3, 4, and 5). Submit the following tables: Table 1. Total drag force for each near-wall region turbulence model along with drag force computed using correlation given by Equations (5) and (6). Table 2. Total heat transfer rate for each model along with heat transfer rate computed using correlation given by Equations (8) and (9). Submit answers to the following questions: 1. Comment on your ability to validate your results using the experimental correlations developed for turbulent flow. 2. Comment on what is a reasonable height for the domain for an accurate solution. 3. Comment on what is a reasonable growth factor for an accurate solution. Refer to each of the Figures and Tables produced above as part of your answer. Limit your comments to 1 page of text.

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References [1] Edwards, A. and Furber, B. N., “The Influence of Free-Stream Turbulence on Heat Transfer by Convection from an Isolated Region of a Plane Surface in Parallel Air Flow,” Proc. Inst. of Mechanical Engineers, 170, 1956. [2] Reynolds, W. C., Kays, W. M., Kline, S. J., “Heat Transfer in the Turbulent Incompressible Boundary Layer,” NASA Memo 12-1-58W, December 1958. [3] Rubesin, M., “The Effect of an Arbitrary Temperature Variation Along a Flat Plate on the Convective Heat Transfer in an Incompressible Turbulent Boundary Layer,” NACA Technical Note 2345, 1951. [4] Seban, R. A. and Doughty, D. L., “Heat Transfer to Turbulent Boundary Layers with Variable Freestream Velocity,” Journal of Heat Transfer, 78, 217, 1956. [5] Wang, Q., “On the Prediction of Convective Heat Transfer Coefficients Using General-Purpose CFD Codes,” AIAA Meeting and Exhibit, AIAA 2001-0361, January 2001.

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Appendix A. Replay Script 1. # Replay script for Lab #8 2. # 3. # define parameters 4. # 5. set L 1000 6. set L0 -100 7. set h 50 8. set H 1000 9. set dx 10 10. # 11. # define points 12. # 13. ic_geo_new_family PNT 14. ic_empty_tetin 15. ic_point {} PNT pnt.00 0,0,0 16. ic_point {} PNT pnt.01 $L0,0,0 17. ic_point {} PNT pnt.02 $L,0,0 18. ic_point {} PNT pnt.03 $L,$h,0 19. ic_point {} PNT pnt.04 0,$h,0 20. ic_point {} PNT pnt.05 $L0,$h,0 21. ic_point {} PNT pnt.06 $L0,$H,0 22. ic_point {} PNT pnt.07 0,$H,0 23. ic_point {} PNT pnt.08 $L,$H,0 24. # 25. # define curves with boundary names 26. # 27. ic_geo_new_family WALL 28. ic_curve point WALL crv.00 {pnt.00 pnt.02} 29. ic_geo_new_family INLET 30. ic_curve point INLET crv.01 {pnt.01 pnt.05} 31. ic_curve point INLET crv.02 {pnt.05 pnt.06} 32. ic_geo_new_family OUTLET 33. ic_curve point OUTLET crv.03 {pnt.02 pnt.03} 34. ic_curve point OUTLET crv.04 {pnt.03 pnt.08} 35. ic_geo_new_family SYMMETRY 36. ic_curve point SYMMETRY crv.05 {pnt.01 pnt.00} 37. ic_curve point SYMMETRY crv.06 {pnt.06 pnt.07} 38. ic_curve point SYMMETRY crv.07 {pnt.07 pnt.08} 39. ic_geo_new_family INTERIOR 40. ic_curve point INTERIOR crv.08 {pnt.04 pnt.00} 41. ic_curve point INTERIOR crv.09 {pnt.04 pnt.07} 42. ic_curve point INTERIOR crv.10 {pnt.04 pnt.05} 43. ic_curve point INTERIOR crv.11 {pnt.04 pnt.03} 44. # 45. # define surfaces with zone name 46. # 47. ic_geo_new_family FLUID 48. ic_surface 2-4crvs FLUID srf.00 {0.01 {crv.00 crv.03 crv.08 crv.11}} 49. ic_surface 2-4crvs FLUID srf.01 {0.01 {crv.01 crv.05 crv.08 crv.10}} 50. ic_surface 2-4crvs FLUID srf.02 {0.01 {crv.02 crv.06 crv.09 crv.10}} 51. ic_surface 2-4crvs FLUID srf.03 {0.01 {crv.04 crv.07 crv.09 crv.11}} 52. # 53. # create blocks and autoassociate to overlapping geometry 54. # 55. ic_hex_initialize_mesh 2d new_numbering new_blocking FLUID 56. ic_hex_split_grid 11 13 FLUID.19 m FLUID VORFN 57. ic_hex_split_grid 11 19 FLUID.14 m FLUID VORFN 58. ic_hex_auto_associate_geometry 59. # 60. # set parameters for mesh on block edges 61. # 62. ic_hex_set_mesh 11 37 n 11 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 0 uniform copy_to_parallel unlocked 63. ic_hex_set_mesh 37 19 n 101 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 0 uniform copy_to_parallel unlocked 64. ic_hex_set_mesh 11 33 n 22 h1rel 0.004 h2rel 0.0 r1 2 r2 2 lmax 0 geo1 copy_to_parallel unlocked 65. ic_hex_set_mesh 33 13 n [expr int(($H-$h)/$dx + 1)] h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 0 uniform

copy_to_parallel unlocked

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66. # 67. # create mesh for blocks 68. # 69. ic_hex_create_mesh FLUID proj 2 dim_to_mesh 3 70. # 71. # set boundary conditions 72. # 73. ic_boco_set INTERIOR {{1 INTER 0}} 74. ic_boco_set WALL {{1 WALL 0}} 75. ic_boco_set INLET {{1 VELI 0}} 76. ic_boco_set OUTLET {{1 PRESO 0}} 77. ic_boco_set SYMMETRY {{1 SYM 0}}