Turbulent flows are significantly affected by the presence of walls. Obviously, the mean velocity field is affected through the no-slip condition that has to be satisfied at the wall.
However, the turbulence is also changed by the presence of the wall in non-trivial ways.
close to the wall the flow is influenced by viscous effects and does not depend on free stream parameters. the mean flow velocity only depends on the distanceyfrom the wall, fluid densityρ and viscosityµand the wall shear stress τw [4]. So
U =f(y, ρ, µ, τw)
Formula Eq. (73) is called thelaw of the wall and contains the definitions of two impor-tant dimensionless groups,u+ andy+. Here uτ =p
τw/ρis called friction velocity.
The κ−models, the RSM, and the LES model are primarily valid for turbulent core flows and will not predict correct near-wall behavior if integrated down to the wall.
Therefore, it is necessary to make these models suitable for wall-bounded flows. The Spalart-Allmaras andκ−ωmodels were designed to be applied throughout the boundary layer, provided that the near-wall mesh resolution is sufficient.
Numerous experiments have shown that the near-wall region can be largely subdivided into three layers.
1. Linear or viscous sub-layer:- the fluid layer in contact with a smooth wall At the solid surface the fluid is stationary. Turbulent eddying motions must stop very close to the wall and the behavior of the fluid closest to the wall is dominated by viscous effects. The viscous sub-layer is in practice extremely thin (y+ < 5) and assume that the shear stress is approximately constant and equal to the wall shear stress τw. After some simple algebra and making use of the definitions ofu+ andy+ this lead to
u+=y+ (74)
Because of the linear relationship between velocity and distance from the wall the fluid layer adjacent to the wall is also known as the linear sub-layer.
2. Log-law layer:- the turbulent region close to a smooth wall
Outside the viscous sublayer a region exists where viscous and turbulent effects are both importance. The shear stressτ varies slowly with distance from the wall [4]. and
within this inner region it is assumed to be constant and equal to the wall shear stress.
Relationship betweenu+ and y+ that is dimensionally correct:
u+= 1
kln(y+) +B = 1
kln(Ey+) (75)
Here, von karman’s constant k = 0.4 and the additive constant B = 5.5 or ( E = 9.8) for smooth wall, wall roughness cause a decrease in the value of B. The value ofkand B are universal constants valid for all turbulent flows past smooth walls at high Reynolds number. Formula (75) is often called the log-law, and the layer where y+ takes values between 30 and 500 thelog-law layer.
3. outer layer:- the inertia-dominated region far from the wall
Experimental measurements show that the log-law is valid in the region0.02< y/δ <0.2.
For larger values ofythe velocity-defect law provides the correct form [8]. In the overlap region the log-law and velocity-defect law have to equal and overlap is obtained by assuming the following logarithmic form:
Umax−U ut =−1
klny δ
+A (76)
WhereAis a constant. The velocity-defect law is often called thelaw of the wake.
Figure 6: Subdivisions of the Near-Wall Region [8]
From Fig.6 we can say that the turbulent boundary layer adjacent to a solid surface is composed of two regions [1]:
• The inner region: 10−20%of the total thickness of the wall layer; the shear stress
is constant and equal to the wall shear stressτw. Within this region there are three zones.
1. the linear sub-layer: viscous stresses dominate the flow adjacent to surface.
2. the buffer layer: viscous and turbulent stresses are of similar magnitude.
3. the log-law layer: turbulent stresses dominate.
• The outer region or law-of-the-wake layer: inertia-dominated core flow far from wall; free from direct viscous effects.
4 Flow past a circular cylinder
Flow past a circular cylinder has been the subject of both experimental and numerical studies for decades. This flow is very sensitive to the changes of Reynolds number, a dimensionless parameter representing the ratio of inertia force to viscous force in a flow.
Work in this chapter aims to validate and identify suitable turbulence models in the application of the flow past a circular cylinder. Flow around a circular cylinder has been chosen as pilot study for the investigation on the flow around a bridge deck section due to the effect of vortex shedding on such structures. To begin with, the basic overview of the flow around a circular cylinder and the flow characteristics such as the Strouhal number, vortex shedding, drag, lift, and pressure coefficients are introduced.
4.1 Conceptual overview of flow past a circular cylinder
Flow past a circular cylinder tends to follow the shape of the body provided that the velocity of the flow is very slow, this is known as laminar flow. Flow at the inner part of the boundary layers travels more slowly than the flow near to the free stream. As the speed of the flow increases, separation of flow occurs at some point along the circular cylinder due to the occurrence of the adverse pressure gradient region . Flow separation tends to roll up the flow into swirling eddies, resulting in alternate shedding of vortices in the wake region of the body known as the von Karman vortex street.
4.1.1 Reynolds number
Flow past a circular cylinder varies with the Reynolds number. Small Reynolds number corresponds to slow viscous flow where frictional forces are dominant. When Reynolds number increases, flows are characterised by rapid regions of velocity variation and the occurrence of vortices and turbulence. Mathematically, Reynolds number of the flow around a circular cylinder is represented by,
Re= ρuD
µ (77)
where D is the diameter of the cylinder, u is the inlet velocity of the flow,ρis a density of fluid andµis the dynamic viscosity of fluid.
Experimental study of the flow around a circular cylinder has identified regions where significant patterns of flow occur as the Reynolds number changes, especially when the flow changes from laminar to turbulent state. Generally, the following regimes have been identified from experiment [7]:
Stable range 40< Re <150 Transition range 150< Re <300
Irregular range 300< Re <200,000
Flow becomes very irregular with instabilities beyond Reynolds number of 200,000. An-other dominant feature of the flow around a circular cylinder is the three-dimensional nature of the flow. Bloor (1964) [11] investigated the flow around a circular cylinder between Reynolds number of 200 to 400 when turbulent motion starts to develop in the wake region of the flow. He observed that the transition of flow in the wake region is triggered by large-scale three-dimensional structures.
4.1.2 Vortex shedding and Strouhal number
The separation of flow past a circular cylinder causes pairs of eddies to form alternately on the top and bottom part of the cylinder and travel into the wake region resulting in vortex shedding. Vortex shedding is very common in engineering applications. Figure 7 shows vortex shedding phenomenon in the wake region of the flow past a circular cylinder. Laminar vortex shedding known as the von Karman vortex street has been observed in the wake region of the flow past a circular cylinder at low Reynolds number between 40 to 250. For Reynolds number that is greater than 250, the laminar periodic wake becomes unstable and the eddies start to become turbulent. Further increase of Reynolds number turns the wake region into turbulent flow. Within certain range of Reynolds number (250 < Re < 10,000), the frequency at which vortices are shed in the flow around a circular cylinder tends to remain almost constant.
Figure 7: Vortex shedding in the wake region of the flow past a circular cylinder [31].
Strouhal number is a dimensionless parameter which describes the shedding of the vor-tices in the wake region of a flow. It relates the frequency of vortex shedding to the incident wind speed,
St= fsD
u (78)
where D is the diameter of the cylinder and u is the velocity of the flow. fs is the shedding frequency of vortices equal to 1/T and fs is called the Strouhal frequency or thevortex shedding frequency whereT denotes for period.
4.1.3 Drag, lift and pressure coefficients
The drag force acts in a direction that is opposite of the relative flow velocity (i.e., it opposes the relative flow). It depends on shape and orientation of a body. Drag coefficient,Cd is calculated as follow:
Cd= Fd
1/2ρu2A (79)
where A is the projected area in the flow direction and Fd is the sum of the pressure force and the viscous force components on the cylinder surface acting in the horizontal direction. Lift coefficient,Clis calculated similarly but vertical force is considered rather than horizontal force which is shown in Figure 8.
Cl= Fl
1/2ρu2A (80)
The Strouhal number is related to the Cd of the flow. In sub-critical Reynolds number region (100 < Re < 10,000), increase in the Strouhal number is generally accompanied by a decrease of theCd.
Figure 8: Diagram of forces acting around a circular cylinder.
Theoretically, the drag force is changing at twice the frequency of the lift force for the flow past a circular cylinder or generally flow involving separation. When a vortex is shed from the top of the cylinder, a suction area is created and the cylinder experiences lift. Half a cycle later, an alternate vortex is created at the bottom part of the cylinder. Throughout the process, the lift force changes alternately in a complete cycle of vortex shedding but
the cylinder experiences drag constantly. It is important that any turbulence models can simulate all the forces coefficient correctly for the analysis of the flow past circular cylinder.
Apart from the Cd and Cl, the pressure coefficient,Cp, distribution around the surface of the cylinder is important. Near to the surface of the cylinder, flow momentum is quite low due to viscous effects and thus is sensitive to the changes of the pressure gradient.
Cp plot of the flow around a circular cylinder starting from the stagnation point (zero velocity) whereCp takes a value of one according to Bernoulli’s Theorem. The flow speed then starts to increase accompanied by a drop in the Cp to a negative value.
The speed of flow then starts to reduce near θ = 80◦, [7] along with an increment of pressure in the direction of the flow, which results in the adverse pressure gradient situation. The flow now has to move against the pressure force in addition to the viscous force. This leads to a reduced gradient of the velocity profile and the wall shear stress.
Separation of flow occurs when the shear stress cannot overcome the adverse pressure gradient, this usually happens at 80◦ < θ < 120◦ [7] for sub-critical flow. After the separation point, pressure remains fairly constant in the wake. Accurate prediction of Cp distribution around a cylinder means a turbulence model performs well at predicting the flow separation