Control of the Rough Wall Turbulent Boundary Layer

Introduction

Turbulence is encountered in the various fluid dynamics operations. It is a very undesirable phenomenon that directly affects efficiency. Turbulence is the introduction of the irregularities in the airflow as well as the pressure distribution. This induces the skin friction drag which is known to significantly reduce the efficiency of the device. Turbulence cannot be dismissed completely, it can only be minimized up to a certain amount. Considering the scope in this area, lot of research is conducted in this field. It is very important to minimize the amount of turbulence as much as possible and it can be achieved by modelling the airflow around the device. The process of modelling the airflow around the device to get maximum efficiency is called turbulence modelling. Turbulence modelling is more practiced in the aerospace and automotive sector. This is very important to model a flow around the cars, airplanes to limit the usage of fuel.

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The region closer to the outer surface of the aerodynamic device is considered to be the boundary layer. The boundary layer plays a big part in the behaviour of the turbulence. The boundary layer has a region where the flow is in line or laminar, which is a laminar boundary layer. The region where the flow is disturbed and not linear is called a turbulent boundary layer. The primary reason to study a boundary layer theory is to find the friction drag. The friction drag is calculated by evaluating the pattern of the shear stress distribution over the surface of the wall. Finding the shear stress distribution, velocity profile, and the thickness of the boundary layer are essential for controlling the boundary layer. These terms are calculated for different flows; as mentioned by Hibbeler (2017), for laminar flows by using the Blasius approach and for Turbulent flows by using Prandtl’s one-seventh power law along with a formulation by Prandtl’s and Blasius.

Turbulence modelling is considered to be one of the most difficult topics to study due to its math-heavy basics as well as its dependency on the numerical methods. It may not provide the exact results. It is very important to explore the basic structures before moving to advanced complexities. Considering the difficulties more attention is given to the flow over basics structure, with the less complex flow. Significant research has been conducted on the turbulent boundary layers over a smooth wall surface in the past few years (Kovasznay 1970; Willmarth 1975; Kline 1978; Cantwell 1981; Sreenivasan 1989; Kline and Robinson 1990). Less research has been conducted on the advanced structures where the skin friction is increased due to the introduction of the roughness on the surface (Raupach, Antonia & Rajagopalan 1991).

Based on the results one of the most impressive methods proposed for the control of the flow over a smooth wall was the wall suction method. In this method, region on the wall surface where the turbulent boundary layer formed and then subjected to the wall suction using suction blower. The effect has been studied by comparing the results with or without wall suction. The produced results show that the turbulent boundary layer can be relaminarized by using this method. Limited researchers have tried applying these methods (developed for control of the turbulent boundary layer over the smooth wall) to the rough wall to see the effect.

Boundary Layer Theory

Background

At the beginning of the 20th century, the branch fluid mechanics started getting developed in two directions, theoretical hydrodynamics and hydraulics. The study originated based on application of motion equations to ideal, frictionless and non- viscous fluid. This method always questioned the practicality of the experiments. On the other hand, the science of hydraulics based on the practical approach (Considering all non-ideal factors) to the problems was developed (Schlichting 1960). At the beginning of the current century Ludwig Prandtl worked on the unification of these two branches, and successfully found the co-relation between these two and updated the field of fluid mechanics with the effective amalgamation of these two streams. 

In 1904 Prandtl presented a paper, “On the Motion of Fluids with Very Little Friction” in a Mathematical Congress held at Heidelberg which is considered to be the first document identifying the boundary layer phenomenon. In his eight pages paper, he talked about the theoretical aspects of the boundary layer (Schlichting 1960; Anderson 2005). He showed that the analysis of the much important viscous flows is possible, with some theoretical considerations and with the help of some simplified experiments, he proved that the flow passing over a rigid body can be divided into two different regions, one is the thin layer which sticks to the surface called as boundary layer and the other region away from the body where the effect of friction can be neglected (Anderson 2005).

In 1914 Prandtl showed the results of his famous spheres experiment which depicted the classification of the boundary layer into laminar and turbulent. This discovery was based on the Reynolds founding’s of the classification of flow in fluid dynamics into laminar and 

Turbulent (Schlichting 1960; Dryden 1995). Von Kármán One of the students of the Prandtl in 1921 proposed his well-known equation which involves integration, claiming the computation (approximate) of the boundary layer from the surface (Dryden 1995). After several years of experiments, W. Tollmein was successful in finding the critical value of the Reynolds number, after which the flow starts the transition from the laminar to turbulent. This theory was later verified by H. L. Dryden.

After 1930 the boundary layer theory got popularity and efforts were made by many researchers from all over the world (England, U.S.A, etc.) to understand this important phenomenon directly related to efficiency. At the end of the last century, the rate of papers published on the boundary layer theory is increased by the factor of two.

Boundary layer

Prandtl (1904), proved that fluid flow around the solid body can be divided into two different regions, first one would be the very thin layer close to the solid body which can be called as the boundary layer. The second one would be the region outside this particular layer. In the first region, the friction is considered as its around the surface of the solid body. The second region friction is neglected considering the large distance from the outer surface of the solid body.

Figure 1-1. Boundary-Layer Formation on Smooth Wall (Schlichting 1960).

Figure-1 shows the side view of the boundary layer, where Uꝏ is defined as the free stream velocity. Every fluid particle in the given flow will have its velocity. As the flow proceeds in the X- the direction from the Y-Axis the velocity of the fluid particle increases from the smallest value (Which is 0), when the value of the fluid particle becomes 99% of the free stream velocity that particular point is marked. When all suitable points are marked and drawn in the X-direction, the formed line is called a boundary layer, in figure-1 the boundary line is shown by region below the dotted line. The dotted line is called to be an edge of the boundary layer. δx is defined as the thickness of the boundary layer.

Turbulent Boundary Layer

Figure 1-2. Boundary-Layer Classification (Comsol 2019).

For the external wall-bounded flows, the nearest thin layer to the surface is considered to be a boundary layer. But the boundary layer is a broad term, as there are several parts/regions of the boundary layer. Whenever the fluid passes over the surface (let’s assume the surface as a smooth wall for ease of understanding) up to a certain length, the fluid flow is parallel to the surface in an orderly manner. Eventually, the flow begins to get disturbed and changes to a turbulent flow in which the fluid particles jump from one plane to another. This transition from laminar to turbulent is not sudden. After the demolition of the laminar flow, and before the generation of turbulent flow, a phase occurs where the flow is neither laminar nor turbulent, and this flow/phase is defined to be transitional flow/phase.

In Figure 1-2, the laminar region has particles flow in an orderly manner, and the turbulent region has the fluid particles revolving around their axis. The region in-between the laminar and turbulent is called a transition region where the transition from laminar to turbulent begins. The turbulent boundary layer is the area of interest because it’s the origin of the turbulence. In this work, the objective is to use wall suction just below the turbulent boundary layer region and try to convert the boundary layer into the laminar boundary layer, as much as possible. This can also be called as relaminarization.

The above figure 1-2 is the visual representation of the phases of the boundary layer, but the boundary layer whether it is laminar or turbulent is described using the Reynolds Number. The Reynolds Number is dimensionless quantity gives an idea about the nature of the flow or boundary layer.

Table 1. Boundary-Layer and Reynolds Number (Critical Reynolds Number 2019).

Smooth Wall and Rough Wall

The smooth wall is the flat plate with the plain surface. There are no irregularities introduced. Since the smooth wall comes with no elements that produce roughness on the surface of contact, there is negligible friction with fluid. The rough walls, on the other hand, have irregularities on the surface, which increases the friction and eventually leads to high turbulence. This irregularity could be anything such as cylindrical rods, square-shaped rods, etc. In the last century, a number of researchers has worked on smooth surfaces with very small friction. Rough surfaces were ignored due to the high turbulence introduced by the roughness elements. In which it is very difficult to predict and compute the flow pattern.

The review paper by Raupach, Antonia & Rajagopalan (1991) gives the glimpse of work on turbulent boundary layer in external flows on the smooth wall with a zero pressure gradient (e.g. Kovasznay 1970; Willmarth 1975; Kline 1978; Singh, Radhakrishnan, & Narayan 1988; Antonia, Zhu & Sokolov 1995). The basic turbulence research gives more importance to work on smooth walls, before exploring the advanced complexities such as rough walls or adverse pressure gradient. It is important to study the formation of the boundary layer on smooth surfaces.

The initial hypothesis by Townsend (1976) states that, the outer region of the turbulent boundary layer is the same in both the cases, for a smooth wall as well as for rough wall. The outer region has low shear strength and this is the reason why the outer region is less sensitive to turbulence. Therefore, the outer region is considered to be a region where the turbulence phenomenon is absent. But in the region close to the wall where the shear is large, a significant difference can be observed depending on the smoothness or roughness of the surface. The roughness increases skin friction and alters the structure of the boundary layer (Raupach, Antonia & Rajagopalan 1991).

Krogstad and Antonia (1999) proved that transition from the smooth wall to rough wall causes non- negligible changes in the outer layer. This questions the Townsend (1976) and Raupach, Antonia and Rajagopalan’s (1991) claim on the matter and explains the unpredictability and uncertainty of results in this field of research. Lee and Sung (2007) found the normalization of the turbulent quantities by friction velocities, the roughness introduces the turbulent stresses and vertical turbulent transport in the outer layer (Lee et al. 2009).
 

Control of the turbulent boundary layer

In 1976, Furuya, Miyata, and Fujita investigated the turbulent boundary layer formed on the surface introduced to the roughness in the form of the wires which were placed at equal distance. The effect of this advancement on the flow resistance along the boundary layer was observed. The aluminium plate with the small wires of the cementing elements measuring 2 m long and 1m wide fitted with the right angle to the plate at an equal distance throughout. The whole setup was placed in the wind tunnel and measurements performed using a probe. The pressure distribution around the roughness measured, which revealed that the pressure drag acting on the roughness is a major contributor to the surface roughness and the other frictions were observed to be as same as the smooth wall. 

Antonia, Zhu, and Sokolov (1995) relaminarized the wall using the wall suction through the porous strip method, they observed the sufficiently high suction rate causes the pseudo-laminarization of the boundary layer downstream the strip for a very short distance, up to 70 δ0 (δ0 is the boundary layer thickness at the porous strip). Away from the strip, the boundary layer starts to return to the fully turbulent nature. The skin friction coefficient cf decreases in the value below the value of cf obtained when there was no suction. The relaminarization largely depends upon the Reynolds number/suction rate. The required stream wise distance of the full development of the boundary layer decreases with increasing Reynolds Number (Re) for suction rate (σ). The velocity profiles such as mean and RMS longitudinal apart from the undisturbed profiles of Re and σ. The Skewness and flatness factor dont depend upon Re but subject to changes as per the change in turbulence structure. 

The ‘Direct Numerical Simulation’ method was tested by Lee et al. (2009) to understand the structure of the turbulent boundary layer over a wall roughened by rods. The instantaneous flow field obtained by using the DNS was used to inspect the boundary layer over the given surface. The roughness used was two-dimensional rectangular rods placed equidistant from each other with k/δ = 0.05 (where δ boundary layer thickness). The comparison of characteristics of the turbulent boundary layer over a smooth wall and rough wall gives the details about the effect of surface roughness. Friction velocity is affected with the insertion of the roughness on the smooth wall it has very little effect on outer layer vorticity fluctuations.

Later in 2014, Kamruzzaman et al. changed the roughness elements from rectangular to circular. This study was based on the turbulent boundary layer over a circular bar with two-dimensional transverse. The rod with diameter k was arranged with the same spacing of along the line λ/k of 8 (λ is the distance between two circular bars) which resulted in maximum form drag. For measurements of the mean and fluctuating velocities hot wire anemometry was used, to find out the values of drag. The friction velocity is the measure contributing factor to the roughness effect, there is more importance to this value, and it was measured by using the two methods and the results were compared. The first method was to use the momentum integral equation while the other one was based on measuring the distribution of pressure around the rods. The results obtained from both methods showed consistency for friction velocity to within 3%. Further observation revealed that the drag coefficient is independent of the Re, as it didn’t show the change in static pressure over a change in Re. the displacement height also remained unchanged over this range of Re. the mean velocity showed collapse when scaled with friction velocity and thickness. Which explains that these are the better parameters for the scaling of the rough wall.

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The recent work in this area is performed by Djenidi, Karuzzaman, and Dostal (2019). Where the two-dimensional rough wall turbulent boundary layer was subjected to the wall suction. The Hot- wire anemometry is used for the measurements such as velocity fluctuations. The wall suction was applied to the turbulent boundary layer through a porous strip. The roughness was of the circular rods, placed in the entire length of the wall in the wind tunnel, with a diameter of k = 1.6 mm and were placed uniformly at the distance of 24 mm which is with the ratio λ/k of 15 (this choice of wavelength by k ratio ensures the roughness of the turbulent boundary layer). This showed that after the impact on the roughness element close to the suction strip, the outer part of the boundary layer was diverted towards the inner part. The formation of the vortices due to the introduction of the roughness elements ultimately results in the increase in drag coefficient, which demands more energy at the input or in other words decreases the efficiency significantly. The relaminarization of the boundary layer was not achieved (which was the key point behind this experiment). Also, not much change was observed in the turbulent boundary layer.

Conclusion

Turbulence reduction is desired in the aerospace and automobile industry as it directly affects the efficiency. Since the discovery of a boundary layer theory, much attention has been given to the flows over a smooth surface. In the present literature review, the aim is to understand the concept of the turbulent boundary layer and to study the control strategies developed for the control of the turbulent boundary layer over a smooth wall. The control strategies developed for the smooth wall will be applied to the rough wall to minimize the effect of turbulence.

References

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