Estimation of the dissipation in the roughness sublayer of ...ltces.dem.ist.utl.pt/lxlaser/lxlaser2016/finalworks2016/papers/03.15_… · model, Similarity model or Gradient model,

18th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 – 7, 2016

Estimation of the dissipation in the roughness sublayer of the boundary layer over an Urban-like rough wall using PIV

K. Blackman*, L. Perret, I. Calmet LUNAM Université, École Centrale de Nantes, Nantes Cedex 3, France

* Correspondent author: [email protected]

Keywords: Boundary Layer, Dissipation, Turbulent Kinetic Energy, LE-PIV

ABSTRACT

In the present work a boundary layer developing over a rough-wall consisting of staggered cubes with a plan area packing density λp = 25% is studied within the wind tunnel using Particle Image Velocimetry to investigate the Turbulent Kinetic Energy (TKE) budget. To access the full TKE budget an estimation of the dissipation (ε) using both the transport equation of the resolved-scale kinetic energy and Large-Eddy Particle Image Velocimetry (LE-PIV) models based on the use of a subgrid-scale model following the methodology used in Large-Eddy Simulations. To use these models a low-pass filter must be applied to the data prior to the computation of the velocity gradients in the energy transfer term, using a cut-off wavelength larger than the Taylor micro-scale. This ensures a clear cut-off in the inertial range where the equivalence between the scale energy transfer and the dissipation holds. In the present work a box filter of different size was used within the canopy and above (for heights of z/h ≥ 1). The estimation of the dissipation using the two methods results in an agreement to within 10%, except in the recirculation region within the canopy, which contains small-scales that are not captured using the transport equation. The presence of the cube roughness has a significant influence on the TKE budget due to the strong shear layer that develops over the cubes. The shear layer is shown to produce and dissipate energy, as well as, transport energy through advection, turbulent transport and pressure transport. The recirculation region that forms through interaction of the shear layer and the canopy layer creates rapid longitudinal evolution of the mean flow thereby inducing negative production.

1. Introduction

Turbulent coherent structures in the lower part of the boundary layer developing over urban terrain, such as low and high momentum regions and shear layers induced by the presence of the roughness, have been shown to interact and influence the ventilation of urban canopies (Takimoto et al., 2011; Inagaki et al., 2012). Many studies have focused on identifying these structures and they are well understood qualitatively, but their quantitative relationships are still unknown (Coceal et al., 2007; Volino et al., 2009; Takimoto et al., 2011; Inagaki et al., 2012). This is particularly true with regard to energy transfer and production between turbulent structures, as very few experimental studies in the urban boundary layer have included


significant analysis of the turbulent kinetic energy (TKE or ) budget due to experimental restraints (Castro et al., 2006; Roth et al., 2015). The general form of the TKE budget for a stationary flow is described in Equation 1, where Adv is advection, P is production, T is turbulent transport, Ψ is pressure transport, D is viscous transport and ε is dissipation of TKE.

(1) Typically, experimental methods limit the budget to two-dimensions and exclude the

direct calculation of the dissipation and pressure transport. In previous works the average ε has been estimated using analysis of longitudinal velocity spectra with the residual of the budget being the pressure transport (Castro et al., 2006; Zhu et al., 2007). This method is typically used, as the spatial resolution of experimental methods is normally too coarse to resolve the dissipative scales at which the gradients of ε must be calculated (Sheng et al., 2000). Perhaps due to this limitation, there are few experimental studies of urban boundary layers that include TKE budget calculations in their work (Castro et al., 2006; Roth et al., 2015). Of these studies, Castro et al. (2006) provide the most detailed analysis and include calculations for the production, turbulent transport, advection, ε and pressure transport (estimated from the budget’s residual). The measurements were conducted above a canopy composed of a staggered cube array with a plan area packing density λp = 25%, at a location corresponding to x/h = 0 and y/h = 0 in Fig. 1, where x and y are the streamwise and spanwise location, respectively. The dissipation is estimated from the power spectral density in the inertial range with a maximum error of 20% (Castro et al., 2006). Due to experimental limitations the budget terms are limited to two-dimensional estimates. As expected, the dissipation rate is approximately balanced by the shear production in the inertial layer of the boundary layer. Near the tops of the cubes the turbulent transport term is an energy sink. The residual of the budget is negative and has a large magnitude, which suggests that pressure transport away from the shear layer is significant (Castro et al., 2006). To the authors’ knowledge no other experimental studies have sufficient results for comparison with the present work.

Other methods for quantifying ε have been proposed including estimation from the transport equation of the resolved-scale kinetic energy (Natrajan and Christensen, 2006) and


Large-Eddy Particle Image Velocimetry (LE-PIV) models based on the use of a subgrid-scale model (SGS) following the methodology used in Large-Eddy Simulations (Sheng et al., 2000). This method combines the use of PIV that is unable to resolve the dissipative scales with a SGS model used to model those scales. As in LES, any of the SGS models, such as the Smagorinsky model, Similarity model or Gradient model, can be used in the LE-PIV analysis. However recent studies of smooth-wall boundary layers have demonstrated the importance of using a model, which considers instantaneous dissipative backscatter, such as the Gradient model (Natrajan and Christensen, 2006). Further details on how these models are applied to PIV are provided in Section 3 below.

In the present work the influence of the roughness elements in an urban boundary layer on energy transfer and production will be investigated through quantification of ε and thereby, all the TKE budget terms. The quantification of ε will be performed using both the LE-PIV method proposed by Sheng et al. (2000) and the energy transfer equation (Natrajan and Christensen, 2006).

2. Experimental Details

The experiments were conducted in the low-speed, suction-type boundary layer wind tunnel in the Laboratoire de recherche en Hydrodynamique, Énergétique et Environnement Atmosphérique at Ecole Centrale de Nantes. For a detailed description of the experimental setup and the present dataset the reader is referred to the work of Rivet (2014). The wind tunnel has working section dimensions of 2 m (width) x 2 m (height) x 24 m (length), 5:1 ratio inlet contraction and a wind speed range of 3-10 ms-1. Within the empty wind tunnel there is a free-stream turbulence intensity of 0.5% with spanwise uniformity to within ±5% (Savory et al., 2013). To initiate the boundary layer development five 800 mm vertical tapered spires were used immediately downstream of the contraction followed by a 200 mm solid fence located 750 mm downstream of the spires. The roughness elements, which consisted of 50 mm staggered cubes with a plan area packing density λp = 25%, then followed. The experiment was performed with a free-stream velocity Ue = 5.8 ms-1 measured with a pitot-static tube located 15 m downstream of the inlet at y = 0 m and z = 1.5 m, giving a Reynolds number, based on cube height, of Reh = 1.9x104.

The flow measurements were conducted 19.5 m downstream of the inlet using a Dantec particle image velocimetry system set up in stereoscopic configuration aligned in the streamwise direction to measure all three components of velocity (Fig. 1). Water-glycol droplets generated


using a commercially available smoke generator were used to seed the flow and had a diameter with distribution mean of 1 μm. These seeding particles were introduced just downstream of the contraction of the wind tunnel to ensure proper seeding of the lower part of the boundary layer. A light sheet generated by a Litron double cavity Nd-YAG laser (2 x 200 mJ) was used along with two CCD cameras with a 105 mm objective lens. A frequency of 5 Hz was used between pairs of pulses of the laser and a time-delay of 400 μs was set between two images of the same pair. The synchronization of the cameras and laser was controlled using Dantec Dynamic Studio software, which was also used to perform the PIV analysis of the recorded images. In total 4000 pairs of images were recorded which corresponds to approximately 13 min of measurements. The multi-pass cross-correlation PIV processing resulted in a final interrogation window size of 32 x 32 pixels with an overlap of 50%. Finally, the PIV measurements had a final spatial resolution of 1.7 mm in the streamwise and 2.2 mm in the vertical.

Fig. 1 Wind tunnel stereoscopic PIV set-up of streamwise aligned measurements with PIV

measurement region (____) in the x-z plane (Rivet, 2015)

3. Results 3.1 Dissipation Estimation

As demonstrated latter, the spatial resolution of the present PIV measurements is too

coarse to resolve the dissipative scales at which the gradients in ε (Equation 1) must be calculated. We therefore calculate ε using two methods, direct calculation of the energy transfer in the inertial range of the velocity spectrum (Natrajan and Christen, 2006) and LE-PIV (Sheng et al., 2000) following the LES formalism that assumes the energy transfer rate in the inertial range


from the large to the small-scales is equal to the dissipation. Sheng et al. (2000) proposed to model the transfer rate with the use of a subgrid scale model, as in LES computations. In the present work the gradient model as outlined by Clark et al. (1979) is used to estimate ε of the SGS. The second method, the direct energy transfer calculation outlined by Natrajan and Christensen (2006), depends on the assumption that velocities at different scales produce similar turbulent stresses. The energy transfer method directly calculates the energy transfer between the large-scale filtered turbulence and the small-scale structures. Due to the low spatial resolution of the PIV the small-scale structures in the flow are not fully captured. This method thus may neglect small-scale structures important to energy transfer in urban boundary layers.

The dissipation was determined using Equation 2 where Sij is the resolved strain rate

tensor calculated using Equation 3. τij is the SGS stress tensor and is calculated differently

depending on which method either the LE-PIV gradient model (τijLE-PIV, Equation 4) or direct

energy transfer (τijET , Equation 5) is used. In these equations denotes temporal averaging,

denotes spatial filtering and ∆is the size of the spatial filter used. Finally, in the present work stereoscopic PIV measurements have been performed in one plane making it impossible to compute the spatial gradient of the instantaneous velocity in the three directions of space but allows for computation in the plane of measurement (x-z). For both the direct energy transfer

and LE-PIV methods the missingterm is estimated using the divergence-free condition with

the remaining estimated invoking an isotropy condition (Sheng et al., 2000).

(2)

(3)

(4)

(5)

Estimation of the dissipation using these two methods requires special treatment of the data. The PIV technique acts as a low-pass filtering operation onto the velocity field. However, the size and the type of the filter is not well defined, especially when performing stereoscopic PIV. Moreover, the spatial resolution obtained with the PIV processing algorithm does not necessarily lie in the inertial range of the velocity spectrum, which is required for estimation of the dissipation. Thus, in the present work, a low-pass filter is applied to the data prior to the


computation of the velocity gradients, using a cut-off wavelength larger than the Taylor micro-scale (λ). This ensures a clear cut-off in the inertial range where the equivalence between the scale energy transfer and the dissipation holds (Sheng et al., 2000).

To determine the cut-off wavelength required for the low-pass filter the dissipation was estimated using both the LE-PIV and the energy transfer method with different low-pass filter sizes applied. The dissipation estimate should be independent of the filter cut-off wavelength if this parameter lies in the inertial range of the turbulent spectrum. This will serve here as a criterion to validate the choice of the low-pass filter size. Due to the differences in the nature of the turbulence within the canopy and within the overlying boundary layer two different filter sizes are applied. From Fig. 2 at heights z/h < 1 a filter of size ∆1(x, z) = (28.9 mm, 6.6 mm) must be applied while at heights z/h ≥ 1 a filter of size ∆2(x, z) = (22.1 mm, 6.6 mm) is sufficient. In the present work the same low-pass filter was applied for both the LE-PIV and direct energy transfer estimations. In addition, the Taylor microscale λ is estimated using Equation 6 and compared with low-pass filter size to ensure that the chosen cut-off wavelength of the filters applied is larger than λ. Fig. 3 shows λ along with the low-pass filter size used, the Kolmogorov scale (η) estimated using Equation 7 and the spatial resolution of the PIV (∆x). The estimation of the scales λ and η have been performed using the dissipation from the LE-PIV method. It is clear that the spatial resolution of the PIV is too coarse to capture the scales of the order of η for direct calculation of ε and too fine to fall within the inertial range for calculation of ε using the LE-PIV or energy transfer equations.

(6)

(7)


Fig. 2 Mean dissipation at y/h = 0 and x/h = 0 calculated using direct energy transfer (lines) and LE-PIV gradient model (symbols) with

varying low-pass filter size ∆, all terms normalized by h/u*

3

Fig. 3 Estimation of the Taylor (λ) and Kolmogorov (η) microscales using LE-PIV method showing spatial resolution of PIV

(∆x) and low-pass filter (∆)

The longitudinal spatial spectra computed both from PIV and by invoking Taylor’s

Hypothesis of frozen turbulence with hot-wire anemometry (HWA) measurements taken in the overlying boundary layer can also be used to determine the low-pass filter size and is shown in Fig 4. The spectra within the canopy are unavailable so this method can only be used to confirm the size of ∆2, which is shown to be within the inertial range. The longitudinal spectra are also frequently used to estimate the dissipation (Castro et al., 2006) and in the present work the comparison between the spectral estimate of the dissipation and the LE-PIV and energy transfer methods shows good agree above the canopy at y/h = 0 and x/h = 0 (Fig. 5).

Fig. 4 Longitudinal spectra calculated using PIV (solid lines) and HWA by invoking Taylor’s Hypothesis (dotted lines). Vertical solid black line: low-pass filter size above the canopy (∆2)


Fig. 5 Comparison of dissipation estimates at y/h = 0 and x/h = 0 using longitudinal spectra (☐), LE-PIV gradient model (*) and direct energy transfer, ET (o) methods, all terms normalized

by h/u*3

Fig. 6 shows ε calculated using both the LE-PIV method and the direct energy transfer

method at y/h = 0 and x/h = -0.5, x/h = 0 and x/h = 0.5. The dissipation ε estimated by both models agrees well, to within 10%, within the overlying boundary layer and within the canopy at locations x/h = 0 and x/h = 0.5. However, close to the upstream cube there is a significant discrepancy, up to 40%, between the two models within the shear layer and the canopy. This discrepancy is a result of the low PIV resolution neglecting small-scale structures important to the dissipation within the recirculation region existing in the wake of the upstream cube. Those structures are not taken into account by the direct energy transfer method but their influence is modelled by the LE-PIV model. Therefore, the gradient SGS LE-PIV model will be used in the remaining analysis of the present work.


Fig. 6 Mean dissipation at y/h = 0 and a) x/h = -0.5; b) x/h = 0; c) x/h = 0.5 calculated using

direct energy transfer, ET (o) and LE-PIV gradient model (*) with all terms normalized by h/u*3

3.2 Turbulent Kinetic Energy Budget

Fig. 7 shows the TKE budget at y/h = 0 and x/h = -0.5, x/h = 0 and x/h = 0.5 behind a cube obstacle. Above the roughness sublayer (z/h > 2) the production of energy is balanced by the dissipation, while all other terms are negligible. Close to the obstacles, the strong shear layer induced by the presence of the roughness elements produces energy. This production term decreases in magnitude with downstream distance as the shear layer develops over the canopy. Energy is also dissipated within the shear layer and transported to the canopy and overlying boundary layer through advection, turbulent transport and pressure transport by the shear layer. Finally, within the canopy pressure transport is a significant energy sink and is balanced by production and turbulent transport except close to the upstream cube where pressure transport


becomes an energy source. This region, which comprises a recirculation region, also exhibits high magnitudes of dissipation and negative production.

Fig. 7 Turbulent Kinetic Energy Budget with εLE-PIV at y/h = 0 and a) x/h = -0.5; b) x/h = 0; c) x/h = 0.5 all terms normalized by h/u*

3

The variation of production with x/h, specifically the negative production close to the upstream roughness, requires further examination. Fig. 8 shows the production decomposed into the contribution of each velocity gradient term. At all locations the largest contribution is the term which is a result of the strong shear layer that is induced by the presence of the roughness. However, close to the upstream cube there is a large contribution of the term

which causes negative production in this region. Negative production suggests flow is not in equilibrium and it has been previously shown to occur in asymmetric flows such as, the wake of a thin plate (Kayakawa et al., 1992), the wake of two cylinders of different diameters (Beguier et al., 1978) and around a wall-mounted cube (Yakhot et al., 2006). The


negative production is attributed to a rapid change of the flow in the recirculation region, resulting in positive (Kayakawa et al., 1992; Yakhot et al., 2006). In the present work this rapid increase occurs within the wake of the upstream cube (Fig. 8d) where the shear layer interacts with the canopy layer forming a recirculation region with large positive . Downstream of the wake this term becomes positive reaching a maximum just above the shear layer suggesting that in this region this term is related to the growth of the shear layer.

Fig. 8 Decomposed turbulence production with total production (∆) at y/h = 0 and a) x/h = -0.5; b) x/h = 0; c) x/h = 0.5 all terms normalized by h/u*

3; d) Mean streamwise velocity normalized by Ue

3.3 Influence of Cube Roughness

The influence of the cube roughness on the TKE budget terms is important to understanding energy transfer within the urban boundary layer and Fig. 9 shows the mean contours of each term in the x-z plane. Energy is advected away from upstream cube by the


strong shear layer, while further downstream, as the shear layer develops advection energy is positive. Energy is produced within the shear layer with the magnitude decreasing as the shear layer grows. At the downstream cube there is strong energy production from the formation of a shear layer and separation of the flow. As mentioned above, the shear layer that forms along the upstream cube causes a rapid change of streamwise velocity magnitude and recirculation in the wake of the cube resulting in negative production. As well, it is clear that the shear layer acts to transport energy away through turbulent transport to the canopy. Dissipation is highest in regions where small-scale structures control the flow dynamics, such as the shear layer and the wake of the upstream cube. As the shear layer grows the size of the structures increase and less energy is produced and dissipated. The recirculation region forms a region of positive pressure transport, whereas the shear layer and the flow separation on the downstream cube result in negative pressure transport.

3.4 Comparison with Literature

There are few experimental studies that can be used for comparison with the present work. Castro et al. (2006) provide the TKE budget of a 25% staggered cube array where ε is estimated using the longitudinal velocity spectra. Fig. 10 shows the TKE budget terms of the present work with those of Castro et al. (2006) at y/h = 0 and x/h = 0 (only production, ε and pressure transport are shown for clarity). There is good agreement for the production, however, the present work results in lower magnitude for ε throughout the boundary layer. Obtained estimates agree well with those from Leonardi and Castro (2010) who performed a DNS of cube roughness. These authors stated that unlike smooth wall boundary layers the ratio of production to ε is significantly higher than unity especially near the cubes. Spatial averages show that just above the cubes within the shear layer the ratio of production to ε is approximately 2.8, while in the present work a spatial average in the x-direction results in a ratio of approximately 2.7. Therefore, the discrepancy between the present work and that of Castro et al. (2006) suggests that there is an over estimation of the dissipation as the ratio of production to dissipation in Castro et al. (2006) is close to unity. This might be due to the limitation of the method based on the use of temporal spectrum combined with the Taylor’s hypothesis for estimating ε and demonstrates the benefit from using the present approach to obtain a more accurate dissipation rate.


Fig. 9 Spatial variability of TKE budget terms a) advection; b) production; c) turbulent transport; d) dissipation, all terms normalized by h/u*

3; e) pressure transport.


Fig. 10 Production, ε and pressure transport of turbulent kinetic energy budget compared with results from Castro et al. (2006) at y/h = 0 and x/h = 0, all terms normalized byh/u*

3

4. Conclusions Within the urban boundary layer few experimental studies have attempted to calculate

the turbulent kinetic energy (TKE) budget, particularly ε, which is typically estimated using longitudinal spectra (Castro et al., 2006; Zhu et al., 2007; Roth et al., 2015). In the present work ε is estimated using two methods, an LE-PIV gradient model (Sheng et al., 2000) and a direct energy transfer calculation (Natrajan and Christensen, 2006). Both methods estimate ε using a low-pass spatial filter with a cut-off wavelength larger than the Taylor microscale ensuring that the separation between the large-scales and the small-scales occurs within the inertial range of the velocity spectra (Sheng et al., 2000). Results from the two methods show good agreement, to within 10%, but large discrepancy is evident within the recirculation region close to the upstream cube. This region contains small-scale structures important to the energy transfer that are not fully captured by the PIV, leading to an underestimation of the dissipation rate by the direct energy transfer method. The shear layer induced by the presence of the roughness is shown to produce and dissipate energy, as well as, transport energy through advection, turbulent transport and pressure transport. As the shear layer develops over the canopy the magnitudes of the production and dissipation of energy decrease until separation occurs at the downstream cube causing strong production. A recirculation region that forms within the wake of the


upstream cube through interaction of the shear layer and the canopy layer creates positive pressure transport and contains small-scales structures that dissipate energy. Furthermore, this region results in a rapid increase of the streamwise velocity component in the longitudinal direction thereby creating negative production. The present work will form a basis for future work investigating coherent structures involved in turbulent kinetic energy processes and energy transfer between scales.

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Roth M, Inagaki A, Sugawara H, Kanda M (2015) Small-scale spatial variability of turbulence statistics, (co)spectra and turbulent kinetic energy measured over a regular array of cube roughness. Environ Fluid Mech 15:329-348. Savory E, Perret L, Rivet C (2013) Modelling considerations for examining the mean and unsteady flow in a simple urban-type street canyon. Meterol and Atmos Phys 121:1-16. Sheng J, Meng H, Fox RO (2000) A large eddy PIV method for turbulence dissipation rate estimation. Chem Eng Sci 55:4423-4434. Takimoto H, Sato A, Barlow JF, Moriwaki R, Inagaki A, Onomura S, Kanda M (2011) Particle Image Velocimetry measurements of turbulent flow within outdoor and indoor urban scale models and flushing motions in urban canopy layers. Bound Layer Meteorol 140:295-314. Volino RJ, Schultz MP, Flack KA (2009) Turbulence structure in a boundary layer with two-dimensional roughness. J Fluid Mech 635:75-101. Yakhot A, Liu H, Nikitin N (2006) Turbulent flow around a wall-mounted cube: A direct numerical simulation. Int J Heat Fluid Fl 27:994-1009. Zhu W, van Hout R, Katz J (2007) On the flow structure and turbulence during sweep and ejection events in a wind-tunnel model canopy. Bound Layer Meteorol 124:205-233.

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Estimation of the dissipation in the roughness sublayer of ...ltces.dem.ist.utl.pt/lxlaser/lxlaser2016/finalworks2016/papers/03.15_… · model, Similarity model or Gradient model,