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Publication on energy dissipation in boundary layers when subjected to high levels of free stream turbulence

Publication on energy dissipation in boundary layers when subjected to high levels of free stream turbulence

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    • Fluid Dynamics Research 39 (2007) 305 – 319 Enhanced energy dissipation rates in laminar boundary layers subjected to elevated levels of freestream turbulence Domhnaill Hernon, Ed. J. Walsh∗ Stokes Research Institute, Department of Mechanical and Aeronautical Engineering, University of Limerick, Plassey Technological Park, Limerick, Ireland Received 23 September 2005; received in revised form 18 May 2006; accepted 7 July 2006 Communicated by Y. Tsuji Abstract Enhanced energy dissipation rates in laminar boundary layers subjected to elevated levels of freestream turbulence are investigated for zero pressure gradient flow with freestream turbulence intensities ranging from 0.2% to 7%. The freestream turbulence markedly changes both the mean and fluctuating velocity distributions resulting in increased energy dissipation rates per unit area by up to 34%. A shortcoming of current numerical and analytical techniques is the inability to accurately predict this increased energy dissipation. A new correlation, based upon experimental measurements using the hotwire technique, has been developed to account for this increased rate of energy dissipation. The correlation developed implements the momentum thickness Reynolds number and turbulence intensity at the leading edge to capture the enhanced energy dissipation rates due to elevated freestream turbulence intensity in the laminar boundary layers investigated. The correlation is then applied to well-known flat plate test cases and good agreement is found. © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Energy dissipation rate; Laminar boundary layer; Freestream turbulence; Grid generated turbulence; Thermodynamic boundary layer loss; Laminar loss correlation; Dissipation coefficient ∗ Corresponding author. E-mail address: edmond.walsh@ul.ie (Ed. J. Walsh). 0169-5983/$32.00 © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2006.07.001
    • 306 D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 1. Introduction The influence of freestream turbulence (FST) on boundary layer properties has been investigated ex- tensively over the last number of decades. The majority of work was concerned with the effect of FST intensity and scale on transition onset and length. For turbulence intensities greater than 1% the mode of transition is termed bypass, due to the bypassing of the Tollmien–Schlichting instability. The rapid formation of turbulent spots is the first clear indication that the bypass transition process is underway and a number of studies implementing experimental techniques (Matsubara and Alfredsson, 2001; Fransson et al., 2005), theoretical techniques (Andersson et al., 1999; Luchini, 2000) and direct numerical simulation (DNS) (Jacobs and Durbin, 2001; Brandt et al., 2004) have been carried out in order to elucidate the mechanisms by which a laminar boundary layer undergoes bypass transition. These studies have demon- strated that the initial stages of the bypass transition process contain elongated streaky structures of both positive and negative perturbation velocity which can develop a streamwise waviness and breakdown into turbulent spots via varicose or sinuous secondary instability. The amplitude of these streaky structures grows in proportion to the square root of the streamwise distance and this maximum algebraic growth has been accurately predicted by Andersson et al. (1999) and Luchini (2000). Recently, increased insight into the transition process under elevated FST conditions has been gained through the DNS studies of Jacobs and Durbin (2001) and Brandt et al. (2004) where it was shown that transition occurs on the low-speed streaks that lift up to the upper portion of the boundary layer where they couple with high-frequency disturbances from the freestream. Through these investigations into the bypass transition process came the observation that under the influence of increased FST a laminar velocity profile deviates substantially from the theoretical velocity profiles of Blasius and Pohlhausen. This characteristic has been observed by many investigators, both experimentally by Fasihfar and Johnson (1992), Roach and Brierley (2000), Matsubara and Alfredsson (2001), and numerically by Peneau et al. (2000), Jacobs and Durbin (2001). Although this deviation in velocity gradient has been noted previously, the associated increase in energy dissipation due to enhanced viscous shear rates has not been quantified. This departure in velocity profile compared to the well-established theories of Blasius and Pohlhausen has a two-fold effect on the velocity gradients of the flow: (1) a reduction in the velocity gradient in the outer layer region occurs, increasing the boundary layer thickness compared to theory; (2) an increase in the velocity gradient in the near-wall region develops causing an increase in the wall shear stress with an attendant increase in the rate of energy dissipation. The increase in wall shear stress can be attributed to the enhanced mixing caused by the FST that has been shown experimentally by Volino and Simon (2000) and numerically by Jacobs and Durbin (2001) to penetrate into the boundary layer. The effect of increasing the near-wall velocity gradient has significantly more influence on the energy dissipation rate as the majority of energy dissipation takes place in the near-wall region. One of the most recent analytical attempts at predicting the influence of FST on laminar boundary layer properties was made by Roach and Brierley (2000). They found that the FST and dissipation length scale at the leading edge are critical in determining the characteristics of the laminar regime. Their analytical technique did predict the aforementioned deviation in laminar velocity profiles compared to Blasius; however, they noted that agreement between prediction and experiment for wall shear stress in the laminar regime was poor. Such discrepancies would be amplified when calculating increased energy dissipation as the volumetric energy dissipation rate is proportional to the square of the velocity gradient. Mayle and Schulz (1996) also developed a predictive technique that demonstrated the importance of boundary layer
    • D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 307 fluctuations in defining the physical attributes of pre-transitional flow. However, in unaccelerated flow with slowly decaying FST, Mayle and Schulz (1996) state that their theory predicts the mean velocity profiles to be similar to those of Blasius. From a numerical point of view both LES, (Peneau et al., 2000), and DNS, (Jacobs and Durbin, 2001), have been applied to investigate the influence of increased FST on boundary layer parameters. Both investigations show a considerable increase in laminar skin friction coefficient (Cf ) compared to Blasius theory, which is in qualitative agreement with experiments. However, as stated by Jacobs and Durbin (2001), the magnitude of Cf computed using DNS for the high turbulence intensity flat plate T3B test condition is significantly higher than the Roach and Brierley (1990) data on which the numerical boundary conditions are based. Furthermore, as noted by Volino and Simon (2000), another difficulty with computationally intensive techniques such as DNS is the fact that they may not be practical as a design tool for a number of years, and this still remains the case. Recent investigations such as Roach and Brierley (2000), Jonas et al. (2000) and Fransson et al. (2005) have shown the importance of accurate determination of the FST scales and energy spectra in the laminar boundary layer receptivity process and also in the effect these have on transition onset. For this reason, accurate definition of FST parameters is essential. The length scale of the FST has also been shown to drastically influence the location of transition onset (Jonas et al., 2000; Brandt et al., 2004). However, both of these investigations which incorporated large variations in length scale at constant turbulence intensity showed no increase in laminar Cf thereby indicating that variation in length scale will not cause an increase in laminar energy dissipation. Within the literature little information is available to account for the enhanced energy dissipation with increased FST. This is reflected in both numerical and approximate techniques failing to account for this effect. To this end, the purpose of the current work is to quantify the increased energy dissipation rate in laminar boundary layers due to elevated FST and to develop a correlation that accurately predicts this trend by employing minimum a priori knowledge, thus facilitating relatively simple implementation into commercially available CFD codes. The developed correlation is then used to predict the enhanced energy dissipation rates due to elevated FST for the well-known flat plate T3A and T3B test cases, with good agreement demonstrated. 2. Experimental facility and measurement techniques 2.1. Experimental facility All measurements were obtained in a non-return wind tunnel with continuous airflow supplied by a centrifugal fan. Maximum velocities in excess of 100 m/s can be achieved. The settling chamber consists of honeycomb and wire gauze grids which enable the reduction of flow disturbances generated by the fan. Using hotwire anemometry, low-pass filtered at 3.8 kHz, the background turbulence intensity in the working section of the tunnel was measured at 0.2%. The test section dimensions are 1 m in length by 0.3 m width and height. Turbulence intensities between 0.45% and 7% can be generated using three different turbulence gen- erating grids. Two of the grids are square-hole perforated plates (PP) and the third grid is a square-mesh array of round wires (SMR). The grids are placed at the test section inlet (Fig. 1). Table 1 gives the grid dimensions and the range of turbulence parameters generated by each grid.
    • 308 D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 Fig. 1. Diagram of experimental set-up (not to scale). See Table 2 for L positioning. Table 1 Geometric description of grids and range of turbulence characteristics available, where and are the dissipation and integral length scales, respectively Parameter PP Grid 1 PP Grid 2 SMR Grid bar width (d) (mm) 7 2.6 0.5 Mesh length (M) (mm) 34 25.2 2.5 %Grid solidity 37 20 36 %Tumin 4 2 0.45 %Tumax 7 4.3 3 min (mm) 2 1.8 0.8 max (mm) 11 11 10 min (mm) 9 5 1 max (mm) 14 8.5 3.7 All grids were designed and qualified according to the criteria of Roach (1987). The plate leading edge was always placed at least 10 mesh lengths downstream of the grid and the isotropy of the FST was validated against the von Kármán one-dimensional isotropic approximation given by Hinze (1975), with excellent agreement. The turbulence decay rate for these grids compares favourably to the power law relation of Roach (1987) where the percentage turbulence intensity decays to the power of − 5 . 7 The test surface for the current measurements is a flat plate manufactured from 10 mm thick aluminium approximately 1 m long by 0.295 m wide and is placed in the centre of the test section. The leading edge is semi-cylindrical and 1 mm in radius. The flow over the flat plate was qualified as two-dimensional over all measurement planes. The design of the trailing edge flap was shown to anchor the stagnation streamline on the upper test surface thus allowing for zero-pressure gradient to be established facilitating excellent comparison against Blasius theory. The effectiveness of the leading edge design is obvious when considering that the bulk pressure distribution along the length of the plate varies no more than ±1% except for the most upstream static pressure point, located 30 mm downstream of the leading edge, where a 5% drop in dynamic pressure is measured. Further details on the design, manufacture and characterisation of the turbulence grids and the flat plate can be found in Walsh et al. (2005). See Fig. 1 for experimental arrangement.
    • D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 309 4.92 4.9 4.9 4.88 4.88 4.86 Vs (Volts) 4.84 4.86 4.82 5.9 5.95 6 6.05 6.1 4.84 4.82 0 2 4 6 8 10 t (s) Fig. 2. Hotfilm detection of turbulent spot, U∞ = 17 m/s and %Tu = 1.3. Inlay caption is zoomed in view of the turbulent spot. 2.2. Measurement techniques Mean and fluctuating velocities were measured using an A.A. Lab Systems AN-1005 constant tem- perature anemometer. The hotwire and hotfilm probes were operated at overheat temperatures of 250 and 110 ◦ C, respectively. All measurements were recorded over 10 s periods at a sampling frequency of 10 kHz and were low-pass filtered at 3.8 kHz to eliminate any noise components at higher frequencies. During any boundary layer traverse the temperature in the test section was maintained constant to within ±0.1 ◦ C. Variation in fluid temperature was compensated for by using the technique of Kavence and Oka (1973). The hotwire calibration was obtained using King’s law between test velocities of 0.4 and 20 m/s. As the current investigation focuses on laminar flow, accurate definition of the extent of the laminar regime is necessary, where the downstream limit of the laminar regime is defined as transition onset. Using the method of Ubaldi et al. (1996) the onset of transition was detected using a hotfilm sensor (Dantec 55R47) whereby a turbulent spot was determined due to increased heat transfer from the sensor. Fig. 2 illustrates the increased heat transfer sensed by the hotfilm as a turbulent spot propagates past the sensor. The shape of this turbulent spot is qualitatively similar to that shown in Ubaldi et al. (1996). Also seen in Fig. 2 are the “turbulent looking” fluctuations found in laminar flow mimicking those found in the freestream (Mayle and Schulz, 1996). The onset of transition was determined to occur where one turbulent spot was formed approximately every 10 s, giving an intermittency ( ) of less than 0.5%. Fig. 3 gives confidence to this detection technique where good comparison between transition onset measurements and the well-established correlation of Mayle (1991) under varying turbulence intensity at the leading edge (%Tule ) and momentum thickness Reynolds number (Re ) is evident. This allows the upper limit of the laminar regime to be accurately identified ensuring all measurements obtained were in laminar flow. Measurements in the laminar regime
    • 310 D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 103 Reθ 102 101 1 2 5 10 %Tule Fig. 3. Comparison of transition onset Reynolds numbers, Re , with the transition onset correlations of Mayle (1991) and Fransson et al. (2005). , (Re , %Tule ) = (442.577, 1.3); •, (Re , %Tule ) = (186.209, 3.1); , (Re , %Tule ) = (154.167, 4.2); −5/8 ∗, (Re , %Tule ) = (131.166, 6); , (Re , %Tule ) = (123.139, 7). ——, Mayle (1991) correlation, Re = 400Tule . - - -, Fransson et al. (2005) correlation, Re = 745/Tule , at = 0.1. The subscript le refers to conditions at the leading edge. Table 2 Tabulated results for each measured test condition presented in Fig. 3, where the leading edge distance downstream of the turbulence grid is L −3 −3 %Tule U∞ le (m) × 10 le (m) × 10 L(m) 1.3 17 1.6 1.4 1.3 1.3 15 1.6 1.5 1.3 3.1 6.6 6.4 3.9 3.9 3.1 5.7 6.4 4.2 3.9 4.2 3.4 5.2 4.5 2.6 4.2 2.9 5.2 4.9 2.6 6 2.9 11 6 4.2 6 3.4 11 5.6 4.2 7 2 9.8 6.6 3.4 7 1.8 9.8 7 3.4 were obtained by reducing the transition onset Reynolds number, by repositioning the probe upstream of the transition onset position or by reducing the freestream velocity (U∞ ). Table 2 gives the test conditions presented in Fig. 3. Included in Fig. 3 is the Fransson et al. (2005) correlation which is based on intermittency levels of 10% and relates Re at transition onset to the inverse of Tule , with a constant of 745. The best-fit constant to the current measurements is found to be 700 which is somewhat lower than Fransson et al. (2005); however, this is the expected trend due to the transition
    • D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 311 onset evaluation criteria employed. From Fig. 3 it is clear that a more universal transition model will have to take turbulent length scale effects into account and also account for the intermittency at which measurements of transition onset are defined. Mean and fluctuating streamwise velocity components were measured using a Dantec 55P11 single normal probe. The hotwire probe was connected to a Digiplan Pk 3 stepper motor drive which traversed in 10 m increments. A boundary layer traverse consisting of approximately 60 measurement locations was obtained at each streamwise measurement station with increased resolution in the near-wall region giving sufficient measurement accuracy for calculation of wall shear stress ( w ) and integral parameters. Using the method of Kline and McClintock (1953) the maximum uncertainty in the near-wall velocity used to calculate w was 4% and this resulted in an uncertainty in w of 10%. Based on this near-wall uncertainty in w the maximum uncertainties in the local rate of energy dissipation per unit volume ( ), local rate of energy dissipation per unit area ( ) and the dissipation coefficient (Cd ) were calculated to be 20%, 17% and 18%, respectively. The uncertainty in the boundary layer edge velocity (Ue ), the peak fluctuating streamwise velocity component (urms ) and the momentum thickness ( ) were calculated to be 1%, 3% and 9%, respectively. Any measurement points affected by wall proximity error, which corresponded to a y + value typically less than 5, were deleted from the velocity profiles before data reduction commenced, similar to Roach and Brierley (1990). This approach is also substantiated by the report of McEligot (1985) where it was stated that large errors in w estimation may occur for y + < 5 due to wall/probe interference effects. 3. Theory and method used in the calculation of Cd As stated previously the main objective of the present paper is to quantify and develop a correlation to account for the increased energy dissipation rates in laminar boundary layers due to elevated FST. As stated by Schlichting (1979) the mean value of the dissipation can be assessed through the variation in terms , where is the incompressible viscous dissipation function and is given by 2 2 2 2 2 2 du dv dw du dv dv dw dw du =2 + + + + + + + + . (1) dx dy dz dy dx dz dy dx dz Considering the flow to be two-dimensional, a number of terms in can be neglected, where all contributions to except du/dy are considered negligible (Schlichting, 1979). Therefore, the mean value of the dissipation for a laminar boundary layer can be written as 2 du = . (2) dy The local rate of energy dissipation ( ) is commonly written as (Schlichting, 2000) 2 du = . (3) dy Here , , u, v, w, y, and are the dynamic viscosity, viscous dissipation function, streamwise velocity, wall-normal velocity, spanwise velocity, distance from the wall and fluid density. The over bars represent the time-average quantities.
    • 312 D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 Schlichting (1968) defines the dimensionless friction work performed in the laminar boundary layer by the shearing stress ( ) as d1 + t1 0( / )(du/dy)2 dy 3 = 3 = Cd , (4) Ue Ue where d1 is the portion which is transformed into heat and t1 is the energy of the turbulent motion, which is considered to be negligible when compared to d1 . Eq. (4) is a dimensionless representation of Eq. (3) across the boundary layer thickness and therefore can be considered to be a non-dimensional viscous dissipation per surface area (Cd ) similar to Moore and Moore (1983), Denton (1993) and Hodson and Howell (2005). Moore and Moore (1983) state that Cd relates the dissipation integral or the shear work to the loss of mean kinetic energy and this can be determined by measuring the velocity profile. The term in the integrand is the energy dissipation per unit area (e ) and is the integration of Eq. (3) across the boundary layer thickness ( ). Eq. (5) is the laminar dissipation coefficient obtained by integrating the well-known Pohlhausen velocity profile for zero-pressure gradient flow (Denton, 1993): 0.1746 Cd = . (5) Re The first step in quantifying Cd is to accurately evaluate . Fig. 4(a) illustrates a typical velocity profile in wall coordinates. It is clear that the near-wall measurement resolution is sufficiently high to allow for accurate calculation of w as the velocity profile compares favourably to the linear law of the wall, u+ = y + . The method employed here to evaluate is to utilise the known physical characteristics of laminar boundary layer flow to accurately apply two piecewise curve fits to the velocity profiles. It is well known that w is constant where the velocity profile matches the linear law of the wall and that all velocity profiles must tend towards zero velocity at the wall (Schlichting, 1979). Therefore, according 70 0.5 60 0.4 50 ε'''(m2/s2) 40 0.3 u+ 30 0.2 20 0.1 10 0 0 100 101 102 0 1 2 3 4 5 (a) y+ (b) y (m) x10−3 Fig. 4. Typical velocity profile in wall coordinates compared to the linear law of the wall. , Re =94, %Tule =7; ——, u+ =y + . (b) ——, Local rate of energy dissipation, , calculated for square symbols case in (a).
    • D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 313 to Eq. (3), in this near-wall region must be constant and this constant value can be extrapolated to the wall. By fitting an overlapping sixth-order polynomial to the velocity profile and calculating , from Eq. (3), a second curve can be generated which decreases asymptotically to zero as the boundary layer edge is approached. The area under the resulting curve, given by the full line in Fig. 4(b), determines . 4. Results and discussion 4.1. Characteristics of laminar boundary layers under the influence of elevated FST Hotwire traces for three non-dimensional distances from the wall are shown in Fig. 5(a), where two locations are within a laminar boundary layer and one is in the turbulent freestream. The character of the hotwire signal changes dramatically as the wall is approached, in that the near-wall region is undisturbed by the high-frequency events in the freestream. The corresponding energy spectra are detailed in Fig. 5(b) where the boundary layers receptivity to select frequencies is more evident. From the energy spectra it is seen that the low-frequency content increases and the high-frequency content decreases as the boundary layer is traversed from the freestream to the wall, Matsubara and Alfredsson (2001) report similar findings. The middle velocity trace in Fig. 5(a) is at the location of the peak urms and it is clear from the corresponding energy spectrum that the majority of energy in this region is contained within the lower-frequency range, where the bound- ary layer has selectively damped out the higher frequencies. These trends in the laminar boundary layer receptivity process due to the continuous forcing of the FST are in good agreement with those presented by Volino and Simon (2000) and Matsubara and Alfredsson (2001). Fig. 6(a) presents the streamwise fluctuations measured in the laminar boundary layers investigated. The fluctuating velocity component distributions are normalised with respect to their maxima and the profiles are self-similar throughout the laminar regime, illustrating the Klebanoff mode which is predicted accurately by transient growth theory. The peak fluctuations are found at y/ 1 ≈ 1.4 and decrease to the freestream value at the boundary layer edge. Again these findings are in agreement with Roach and 5 10−2 4 10−4 3 E (m2/s) U (m/s) 10−6 2 10−8 1 0 10−10 0 0 0.05 0.1 0.15 0.2 10 101 102 103 (a) t (s) (b) f (Hz) Fig. 5. (a) Velocity traces, for %Tule of 4.2, at two different heights through the laminar boundary layer and in the freestream. (b) Corresponding energy spectra; - - -, y/ 1 = 0.73; · · ·, 1.42, ——, 12.6.
    • 314 D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 1 1 0.8 urms/u rms,max 0.8 0.6 U/U 0.6 0.4 0.4 0.2 0.2 0 0 0 5 10 0 5 10 15 (a) y/δ1 (b) η Fig. 6. (a) Growth of disturbances through the laminar boundary layers investigated. (b) Deviation in laminar velocity profiles due to increased FST compared to Blasius velocity profile. ◦, (Re , Xle , %Tule ) = (178, 0.255 m, 0.2); , (Re , Xle , %Tule ) = (442, 0.455 m, 1.3); , (Re , Xle , %Tule ) = (209, 0.195 m, 3.1); , (Re , Xle , %Tule ) = (167, 0.272 m, 4.2); •, (Re , Xle , %Tule )=(166, 0.255 m, 6); , (Re , Xle , %Tule )=(139, 0.325 m, 7); ——, Blasius profile. Where Xle denotes the distance of the probe downstream of the plate leading edge. Brierley (1990), Mayle and Schulz (1996) and Matsubara and Alfredsson (2001). It can be seen from Fig. 6(a) that in the outer half of the boundary layer, especially at higher FST, there is significantly increased turbulence activity. This is a characteristic that transient growth theory fails to predict. This increased turbulence activity may be caused by the freestream eddies continuously penetrating the outer portion of the boundary layer thus causing increased mixing in that region (Andersson et al., 1999; Jacobs and Durbin, 2001). Fig. 6(b) illustrates the deviation in the intermittent mean velocity profiles compared to the Bla- sius velocity profile with increased FST. At the lower turbulence intensities, %Tule = 0.2 and 1.3, the profiles compare favourably with the theoretical Blasius profile. With increased turbulence inten- sity there is a marked departure from the Blasius profile whereby increased shear stress at the wall and decreased gradient near the boundary layer edge are observed. In comparable zero-pressure gra- dient flows, Roach and Brierley (1990) reported a similar finding, as too did Matsubara and Alfreds- son (2001). Fig. 6(b) suggests that the magnitude of deviation is proportional to turbulence inten- sity, with higher turbulence intensities causing larger deviation from the theoretical Blasius velocity profile. Fig. 6(a) demonstrates the degree to which streamwise fluctuations are present in a laminar boundary layer under the influence of elevated FST. It is interesting to note that the result of these fluctuations on the instantaneous in the near-wall region is a continuous variation in the magnitude of the dissipation rate. Fig. 7 shows such a plot of at y + ≈ 5. The corresponding variation in y + due to these fluctuations is no more than ±1, hence the linear assumption is assumed to hold. This observation is far removed from the assumed steady laminar profile of many studies and existing prediction techniques. The fluctuations illustrated in Fig. 7 could be attributed to the undulation of streaky structures where increased energy dissipation in the near-wall region may be caused by the displacement of outer-layer high-speed fluid into the near-wall region.
    • D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 315 0.06 0.05 0.04 ε'''(m2/s2) 0.03 0.02 0.01 0 0 0.2 0.4 0.6 0.8 1 t (s) Fig. 7. Instantaneous energy dissipation rate per unit volume at y + ≈ 5 for test conditions presented in Fig. 4. 4.2. Measure of increased energy dissipation rate in laminar boundary layers Fig. 8 demonstrates the variation of the measured laminar Cd with elevated FST compared to the Pohlhausen prediction of Eq. (5). The measurements were taken between 0.2 and 7 %Tule . Also included in Fig. 8 are the flat plate test cases T3A and T3B of Roach and Brierley (1990). At 0.2 %Tule the measured Cd values compare favourably to Eq. (5), with maximum variation of ±2%. This gives confidence in the experimental set-up and also in the technique used to calculate Cd . At the lowest grid generated turbulence intensity of 1.3 %Tule the maximum deviation in Cd is 7%. A similar trend is found with increased turbulence intensity, but with markedly larger deviation compared to Eq. (5). At the highest FST intensity of 7% the maximum deviation is 34%. It can be seen from Fig. 8 that with elevated FST there is significant increase in energy dissipation rates throughout the laminar boundary layers presented. The effect of turbulent length scale on the energy dissipation rates could not be evaluated as the length scales generated during the current set of experiments were approximately equal (Table 2). However, length scale effects are not believed to increase the energy dissipation rate in laminar flow, see Jonas et al. (2000) and Brandt et al. (2004) for examples. 4.3. Correlation of results Fig. 9 illustrates the relationship between the variation in Cd and Eq. (5) for the laminar test conditions measured in the current investigation only. Although a number of methods were investigated to collapse the data, including the use of dissipation and integral length scales, it was found that the combination of %Tule and Re gave the best results. The trend line fit to the data points is exponential and is forced to intersect the ordinate axis at 1. The explanation for this can be seen in Fig. 8 where with the combination of low FST and/or low Re the measured Cd is equal to Eq. (5). The equation for the trend line fit according
    • 316 D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 x 10−3 4 3.5 3 2.5 Cd 2 1.5 1 0.5 0 0 100 200 300 400 500 Reθ Fig. 8. Increase in laminar dissipation coefficient, Cd , with increase in turbulence intensity at leading edge, %Tule .+, (%(Cd /Cd Eq.5 ),max , %Tule )=(34, 7); , (%(Cd /Cd Eq.5 ),max , %Tule )=(32, 6); , (T3B) (%(Cd /Cd Eq.(5) ),max , %Tule ) = (26, 6); , (%(Cd /Cd Eq.(5) ),max , %Tule ) = (22, 3.1); , (T3A), (%(Cd /Cd Eq.(5) ),max , %Tule ) = (16, 3); , (%(Cd /Cd Eq.(5) ),max , %Tule ) = (7, 1.3); ◦, (%(Cd /Cd Eq.(5) ),max , %Tule ) = (2, 0.2); ——, Eq. (5), Cd = 0.1746Re−1 . 2 1.5 (Cd /CdEq. 5 1 0.5 0 0 200 400 600 800 1000 1200 %Tule Reθ Fig. 9. Variation in Cd for the measured test conditions in the current investigation compared to Eq. (5), with variation in %Tule and Re .+, %Tule = 7; , 6; •, 4.2; , 3.1; , 1.3; ——, Exponential trendline given by Eq. (6); - - - -, ±10%.
    • D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 317 to Fig. 9 is given by Cd = exp0.00025%Tule Re . (6) Cd Eq. (5) Hence the correlation to predict the increased energy dissipation in laminar boundary layers under the influence of elevated FST is given by 0.1746 Cd = exp0.00025%Tule Re , (7) Re where the energy dissipation per unit surface area is contained within Eq. (4). The accuracy of the proposed correlation can be judged by the fact that Eq. (6) captures to within ±10% (± two standard deviations, 2 ) all of the data obtained in the current experiment. This comparatively good degree of correlation also substantiates the statement that variation in turbulent length scales does not alter the energy dissipation in the laminar regime. From Fig. 10 it can be seen that with limited information about the flow field, %Tule and Re , an improved prediction of the increased energy dissipation due to elevated FST in a laminar boundary layer may be achieved using the correlation of Eq. (7). Included in Fig. 10 is the predicted increase in Cd for the T3A and T3B test cases. The correlation given by Eq. (7) is shown to predict the increase in Cd to within ±10% for the T3A and T3B test cases. This is a promising result as Eq. (7) is shown to accurately predict the increase in Cd under elevated FST conditions based on measurements with different experimental conditions. The 1.3 and 6 %Tule test cases are included in Fig. 10 for illustration. The sensitivity of the x 10−3 8 6 Cd 4 2 0 0 100 200 300 400 500 Reθ Fig. 10. Comparison between predicted Cd using Eq. (7) and measured Cd due to increase in %Tule , for three conditions. , %Tule = 6; , (T3B) %Tule , =6; , (T3A) %Tule = 3; , %Tule = 1.3; - - - -, Eq. (5), Cd = 0.1746Re−1 ; ——, Correlation presented in Eq. (7).
    • 318 D. Hernon, Ed. J. Walsh / Fluid Dynamics Research 39 (2007) 305 – 319 correlation to variation in the exponent in Eq. (7) was assessed by omitting the extreme data points in Fig. 9 corresponding to the 1.3 and 7 %Tule test cases. This lead to a maximum increase of 3% in the predicted Cd value using Eq. (7). The use of this correlation as a benchmark test for future computations of pre-transitional flow under the influence of elevated FST is also promising as it provides better agreement with measurement than current theories. However, limitations to the proposed correlation exist. In its present state the correlation does not account for streamwise pressure gradient, streamwise curvature or roughness, nor does it take into account heat transfer mechanisms, all of which are important factors in various applications. 5. Conclusions It has been demonstrated that under the influence of elevated FST laminar velocity profiles deviate considerably from the theoretical velocity profiles of Blasius and Pohlhausen. This deviation consists of an increase in the wall shear stress which leads to enhanced energy dissipation rates when compared to theory. To date, quantification, prediction and correlation of this enhanced energy dissipation rate in laminar boundary layers due to elevated FST has not been available. In order to resolve these issues the current investigation has incorporated a broad range of measurement conditions with %Tule ranging between 0.2% and 7% and Re ranging from 60 to 450. It was found that under the influence of FST the energy dissipation rate per unit surface area (measured in terms of the non-dimensional dissipation coefficient Cd ) in the laminar boundary layers investigated increased by 7% at the lowest grid generated turbulence intensity of 1.3%Tule to 34% at the maximum turbulence intensity of 7%Tule . This investigation has provided a new correlation that improves the prediction of energy dissipation rates in the presence of FST in laminar boundary layers. The correlation was applied to predict increased energy dissipation due to elevated FST in the T3A and T3B flat plate test cases and good agreement was found. With limited a priori knowledge of the flow field required, %Tule and Re , this work provides a methodology for improved prediction of energy dissipation rates in laminar boundary layers under the influence of elevated FST from both analytical and numerical perspectives. Acknowledgements This publication has emanated from research conducted with the financial support of science foundation Ireland. The authors also wish to thank the H.T Hallowell Jr Graduate Scholarship for financial assistance during the course of this investigation. References Andersson, P., Breggren, M., Henningson, D.S., 1999. Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150. Brandt, L., Schlatter, P., Henningson, D.S., 2004. Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198. Denton, J.D., 1993. Loss mechanisms in turbomachines. J. Turbomach. 115, 621–654. Fasihfar, A., Johnson, M.W., 1992. An improved boundary layer transition correlation. ASME Paper 92-GT-245.
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