Turbulence Theory & Models

Cheatsheet Content

### Preamble This cheatsheet provides a comprehensive overview of key concepts in Fluid Mechanics and Turbulent Flow, specifically tailored for graduate-level students. Each section covers theoretical derivations, physical interpretations, and practical implications, often concluding with a list of relevant practice questions from the provided Q&A bank. Tensor (Einstein) notation is used throughout. ### Reynolds-Averaged Navier-Stokes (RANS) Equations #### Context The Navier-Stokes equations describe instantaneous turbulent motion. Direct numerical simulation (DNS) of all eddies is computationally prohibitive at high Reynolds numbers. Osborne Reynolds (1895) proposed decomposing the flow into a mean and a fluctuating component, then averaging, leading to the RANS equations. #### Assumptions 1. **Incompressible flow:** $\rho = \text{const.}$ 2. **Newtonian fluid:** Constant dynamic viscosity $\mu$. 3. **No body forces** (or absorbed into pressure). 4. **Reynolds decomposition:** $u_i = \overline{U}_i + u'_i$, $p = \overline{P} + p'$. #### Reynolds Rules of Averaging For two random variables $f, g$ and a constant $c$: - $\overline{f+g} = \overline{f} + \overline{g}$ - $\overline{cf} = c\overline{f}$ - $\overline{\overline{f}} = \overline{f}$ - $\overline{f'} = 0$ - $\overline{\overline{f}g'} = 0$ - $\overline{\frac{\partial f}{\partial x_i}} = \frac{\partial \overline{f}}{\partial x_i}$, $\overline{\frac{\partial f}{\partial t}} = \frac{\partial \overline{f}}{\partial t}$ - **Crucial:** $\overline{fg} \neq \overline{f}\overline{g}$ (generally) #### Derivation Steps **1. Substitute Decomposition into Continuity Equation:** $\frac{\partial (\overline{U}_i + u'_i)}{\partial x_i} = 0 \implies \frac{\partial \overline{U}_i}{\partial x_i} + \frac{\partial u'_i}{\partial x_i} = 0$ Averaging yields $\frac{\partial \overline{U}_i}{\partial x_i} = 0$, and consequently $\frac{\partial u'_i}{\partial x_i} = 0$. Both mean and fluctuating fields are individually solenoidal. **2. Rewrite Convective Term in Conservative Form:** Using continuity, $u_j \frac{\partial u_i}{\partial x_j} = \frac{\partial (u_i u_j)}{\partial x_j}$. **3. Substitute Decomposition and Average:** The product $u_i u_j = (\overline{U}_i + u'_i)(\overline{U}_j + u'_j) = \overline{U}_i \overline{U}_j + \overline{U}_i u'_j + u'_i \overline{U}_j + u'_i u'_j$. Averaging (using $\overline{u'_i} = 0$): $\overline{u_i u_j} = \overline{U}_i \overline{U}_j + \overline{u'_i u'_j}$. **4. Final RANS Equation (Momentum):** Substituting into the instantaneous Navier-Stokes momentum equation and averaging yields: $$\frac{\partial \overline{U}_i}{\partial t} + \overline{U}_j \frac{\partial \overline{U}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \overline{P}}{\partial x_i} + \nu \frac{\partial^2 \overline{U}_i}{\partial x_j \partial x_j} - \frac{\partial (\overline{u'_i u'_j})}{\partial x_j}$$ The new term $-\rho \overline{u'_i u'_j}$ is the **Reynolds stress tensor**, $\tau^R_{ij}$. #### Physical Interpretation The RANS equation is structurally identical to the Navier-Stokes equation, except for the appearance of the Reynolds stress, which represents the additional momentum flux due to turbulent fluctuations. This term introduces 6 new unknowns ($\overline{u'_i u'_j}$) and necessitates closure models. **Relevant Questions:** Q11, Q57, Q58, Q66 ### Reynolds Stress: Definition & Generation Mechanism #### Definition The Reynolds stress tensor $\tau^R_{ij} = -\rho \overline{u'_i u'_j}$ arises from the time-averaging of the convective term in the Navier-Stokes equations. In 3D, it is a symmetric tensor with 6 independent components: $$ \begin{pmatrix} -\rho \overline{u'^2} & -\rho \overline{u'v'} & -\rho \overline{u'w'} \\ -\rho \overline{v'u'} & -\rho \overline{v'^2} & -\rho \overline{v'w'} \\ -\rho \overline{w'u'} & -\rho \overline{w'v'} & -\rho \overline{w'^2} \end{pmatrix} $$ - **Diagonal terms** ($-\rho \overline{u'^2}$) are **normal stresses** (represent turbulent kinetic energy components). - **Off-diagonal terms** ($-\rho \overline{u'_i u'_j}, i \neq j$) are **shear stresses** (represent turbulent momentum transport). #### Physical Interpretation of Reynolds Shear Stress Consider a simple shear flow $\overline{U} = \overline{U}(y)$. The Reynolds shear stress is $-\rho \overline{u'v'}$. - A positive $v' > 0$ (upward fluctuation) carries a fluid parcel from $y_0$ to $y_0 + \delta y$ where the mean velocity is higher. - At $y_0+\delta y$, the parcel still has the slower velocity from $y_0$, so $u' 0$, $v' > 0$ implies $u' 0$ when $\frac{d\overline{U}}{dy} > 0$. This is analogous to viscous stress $\mu \frac{d\overline{U}}{dy}$, but arises from turbulent mixing of momentum, not molecular motion. #### Why $\overline{u'_i} = 0$ but $\overline{u'_i u'_j} \neq 0$ - $\overline{u'_i} = 0$ by definition of Reynolds decomposition (mean of fluctuation is zero by construction). - $\overline{u'_i u'_j} = 0$ would require $u'_i$ and $u'_j$ to be statistically independent (uncorrelated). In shear flows, $u'_i$ and $u'_j$ are strongly correlated. For example, in a flow with $\frac{d\overline{U}}{dy} > 0$, upward moving fluid ($v' > 0$) tends to have lower x-momentum ($u' ### Eddy Viscosity Hypothesis and Closure Problem #### The Closure Problem The closure problem refers to the fundamental mathematical difficulty that arises when the Navier-Stokes equations are averaged: the averaging process always generates more unknowns than equations. - **RANS equations** for 3D flow have 4 equations (1 continuity, 3 momentum) but 10 unknowns ($\overline{U}_1, \overline{U}_2, \overline{U}_3, \overline{P}$ and 6 independent Reynolds stresses $\overline{u'_i u'_j}$). This is 6 equations short. - Deriving transport equations for $\overline{u'_i u'_j}$ introduces third-order triple correlations ($\overline{u'_i u'_j u'_k}$), leading to an infinite hierarchy of moment equations. #### Eddy Viscosity (Boussinesq) Hypothesis Boussinesq (1877) drew an analogy with molecular viscosity: turbulent stresses are proportional to the mean strain rate via a scalar turbulent (eddy) viscosity $\mu_t$: $$\tau^R_{ij} = -\rho \overline{u'_i u'_j} = 2 \mu_t \overline{S}_{ij} - \frac{2}{3} \rho k \delta_{ij}$$ where $\overline{S}_{ij} = \frac{1}{2} \left( \frac{\partial \overline{U}_i}{\partial x_j} + \frac{\partial \overline{U}_j}{\partial x_i} \right)$ is the mean strain-rate tensor and $k = \frac{1}{2} \overline{u'_i u'_i}$ is the turbulent kinetic energy. The term $\frac{2}{3} \rho k \delta_{ij}$ is often absorbed into a modified pressure term. #### How it Closes the System The eddy viscosity hypothesis reduces the 6 unknown Reynolds stress components to one scalar field $\mu_t$, converting the underdetermined RANS system into a closed, solvable set of equations. Different turbulence models then prescribe $\mu_t$ differently. #### Physical Interpretation & Limitations - $\mu_t$ represents the enhanced diffusivity of momentum by turbulent eddies. - **Fundamental limitation:** The hypothesis assumes turbulent stresses are aligned with mean strain rates, equivalent to assuming local isotropy. This fails in flows with strong streamline curvature, swirl, or reattachment, where Reynolds Stress Models (RSM) are necessary. **Relevant Questions:** Q1, Q58 ### Turbulent Kinetic Energy (TKE) Equation #### Definition Turbulent Kinetic Energy (TKE) is defined as $k = \frac{1}{2} \overline{u'_i u'_i}$. It represents the kinetic energy of the fluctuating velocity components. #### Derivation from RANS The TKE equation is derived by multiplying the fluctuating momentum equation by $u'_i$ and averaging. This yields: $$\frac{\partial k}{\partial t} + \overline{U}_j \frac{\partial k}{\partial x_j} = P_k - \epsilon - \frac{\partial}{\partial x_j} \left( \frac{1}{2} \overline{u'_i u'_i u'_j} + \frac{1}{\rho} \overline{p' u'_j} - \nu \frac{\partial k}{\partial x_j} \right)$$ The terms on the RHS are: Production ($P_k$), Dissipation ($\epsilon$), and Diffusion (molecular, turbulent, and pressure). #### Physical Significance of Terms | Term | Symbol / Form | Physical Meaning | | :------------------- | :------------------------------------------ | :-------------------------------------------------------------------------------------------------------------- | | **Material Derivative** | $\frac{Dk}{Dt} = \frac{\partial k}{\partial t} + \overline{U}_j \frac{\partial k}{\partial x_j}$ | Rate of change of $k$ following a fluid parcel: time-rate-of-change at a fixed point plus advection by the mean flow. | | **Production** | $P_k = -\overline{u'_i u'_j} \frac{\partial \overline{U}_i}{\partial x_j}$ | Energy extracted from the mean flow by Reynolds stresses working against mean velocity gradients. Source of TKE. Always positive in shear flows. | | **Viscous Dissipation** | $\epsilon = \nu \overline{\frac{\partial u'_i}{\partial x_j} \frac{\partial u'_i}{\partial x_j}}$ | Irreversible conversion of TKE into heat at Kolmogorov scales, where viscous forces dominate. Pure sink of TKE. Always positive. | | **Viscous Diffusion** | $\frac{\partial}{\partial x_j} \left( \nu \frac{\partial k}{\partial x_j} \right)$ | Molecular diffusion of $k$ down its own gradient. Important only in the viscous sublayer; negligible at high Re. | | **Turbulent Diffusion** | $-\frac{\partial}{\partial x_j} \left( \frac{1}{2} \overline{u'_i u'_i u'_j} \right)$ | Transport of $k$ by turbulent fluctuations (triple velocity correlation). Moves $k$ from high to low regions. | | **Pressure Diffusion** | $-\frac{\partial}{\partial x_j} \left( \frac{1}{\rho} \overline{p' u'_j} \right)$ | Transport of $k$ by pressure fluctuations. Redistributes energy spatially; does not create or destroy $k$. | #### Equilibrium Insight In many flows (e.g., the log-layer of a boundary layer), production $\approx$ dissipation ($P_k \approx \epsilon$), the so-called local equilibrium. This forms the basis of mixing-length and many eddy-viscosity models. **Relevant Questions:** Q17, Q60, Q61 ### k-$\epsilon$ Turbulence Model #### Context The k-$\epsilon$ model is a two-equation eddy-viscosity model, meaning it solves two transport equations (one for TKE $k$ and one for its dissipation rate $\epsilon$) to determine the turbulent viscosity $\mu_t$. #### Modelled $k$ Equation $$\frac{\partial k}{\partial t} + \overline{U}_j \frac{\partial k}{\partial x_j} = \frac{\partial}{\partial x_j} \left( \nu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j} + P_k - \epsilon$$ - **LHS (Material derivative):** Transport of $k$ along mean streamlines. - **Diffusion:** Spatial redistribution of $k$. Molecular term $\nu$ is negligible at high Re; turbulent term $\frac{\mu_t}{\sigma_k}$ dominates. - **$P_k$ (Production):** Energy extracted from mean flow. - **$-\epsilon$ (Dissipation):** Irreversible sink converting TKE to heat. #### Modelled $\epsilon$ Equation $$\frac{\partial \epsilon}{\partial t} + \overline{U}_j \frac{\partial \epsilon}{\partial x_j} = \frac{\partial}{\partial x_j} \left( \nu + \frac{\mu_t}{\sigma_\epsilon} \right) \frac{\partial \epsilon}{\partial x_j} + C_{\epsilon 1} \frac{\epsilon}{k} P_k - C_{\epsilon 2} \frac{\epsilon^2}{k}$$ - This equation provides the length/time scale needed to determine $\mu_t$. #### Turbulent Viscosity The turbulent viscosity is calculated as $\mu_t = \rho C_\mu \frac{k^2}{\epsilon}$. #### Standard Model Constants - $C_{\mu} = 0.09$ - $C_{\epsilon 1} = 1.44$ - $C_{\epsilon 2} = 1.92$ - $\sigma_k = 1.00$ - $\sigma_\epsilon = 1.30$ #### Advantages and Disadvantages (relative to k-$\omega$) - **Near-wall behaviour:** Requires wall-damping functions or wall functions. Inaccurate in viscous sub-layer. - **Free shear flows:** Excellent, well-calibrated. - **Adverse pressure gradient:** Overpredicts turbulent kinetic energy near separation. - **Separated flows:** Under-performs, often delays predicted separation significantly. - **Stability:** Very robust numerically. - **Free-stream sensitivity:** Not sensitive to free-stream $\epsilon$ specification. - **Best use case:** Industrial flows: pipe networks, mixing vessels, far-wake turbomachinery. **Relevant Questions:** Q3, Q14, Q54, Q56 ### k-$\omega$ Turbulence Model #### Context The k-$\omega$ model is another two-equation eddy-viscosity model, solving transport equations for TKE $k$ and specific dissipation rate $\omega$. #### Specific Dissipation Rate ($\omega$) $\omega$ is defined as $\omega = \frac{\epsilon}{k C_\mu}$. It has dimensions of $[s^{-1}]$ and can be interpreted as the rate at which turbulence is being destroyed per unit energy. #### Derivation of $\omega$-Equation (Wilcox Standard Form) $$\frac{\partial \omega}{\partial t} + \overline{U}_j \frac{\partial \omega}{\partial x_j} = \frac{\partial}{\partial x_j} \left( \nu + \frac{\mu_t}{\sigma_\omega} \right) \frac{\partial \omega}{\partial x_j} + \alpha \frac{\omega}{k} P_k - \beta \omega^2$$ - **Production term:** $\alpha \frac{\omega}{k} P_k$ (production of specific dissipation tracks production of $k$). - **Destruction term:** $-\beta \omega^2$ (when eddies break down, the process accelerates). #### Turbulent Viscosity $\mu_t = \rho \frac{k}{\omega}$ (for the k-$\omega$ model). #### Standard Model Constants (Wilcox 2006) - $\alpha = 13/25$ - $\beta = 3/40$ - $\beta^* = 9/100$ - $\sigma_\omega = 0.5$ - $\sigma_k = 0.6$ #### Physical Interpretation The $\omega$ equation tells us: turbulence is being produced at a rate proportional to $\frac{\omega}{k}P_k$ and destroyed quadratically as $\beta \omega^2$. The near-wall behaviour of $\omega \to \infty$ is what gives the k-$\omega$ model its superior near-wall performance without requiring wall-damping functions. #### Advantages and Disadvantages (relative to k-$\epsilon$) - **Near-wall behaviour:** Has an exact analytical near-wall solution ($\omega \sim 6\nu/y^2$). No wall functions needed. - **Free shear flows:** Sensitive to free-stream $\omega$, spurious production. - **Adverse pressure gradient:** Better prediction of separation, less overprediction of $\mu_t$. - **Separated flows:** More accurate for mildly separated flows. - **Stability:** Can be sensitive to initial and BC conditions for $\omega$. - **Free-stream sensitivity:** Strongly sensitive to specified $\omega_\infty$, small changes alter boundary layer significantly. - **Best use case:** Aerodynamic flows with wall-bounded regions, mild separation; baseline for SST. **Relevant Questions:** Q14, Q54, Q55, Q56 ### k-$\omega$ SST Turbulence Closure Model #### Context The Shear Stress Transport (SST) model, developed by Menter (1994), combines the strengths of the k-$\omega$ and k-$\epsilon$ models. - **Superior near-wall accuracy** of the k-$\omega$ model. - **Free-stream insensitivity** of the k-$\epsilon$ model. The key innovation is a **blending function $F_1(x)$** that smoothly transitions between the two models. #### Governing Transport Equations - **k-equation:** Same as standard k-epsilon: $$\frac{\partial k}{\partial t} + \overline{U}_j \frac{\partial k}{\partial x_j} = \frac{\partial}{\partial x_j} \left( \nu + \sigma_k \mu_t \right) \frac{\partial k}{\partial x_j} + P_k - \beta^* k \omega$$ - **$\omega$-equation (blended):** $$\frac{\partial \omega}{\partial t} + \overline{U}_j \frac{\partial \omega}{\partial x_j} = \frac{\partial}{\partial x_j} \left( \nu + \sigma_\omega \mu_t \right) \frac{\partial \omega}{\partial x_j} + \alpha \frac{\omega}{k} P_k - \beta \omega^2 + 2(1-F_1) \sigma_{\omega 2} \frac{1}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$$ The last term is the **cross-diffusion term**, which is active when transforming k-$\epsilon$ to k-$\omega$ form and is significant in the outer region ($F_1 \to 0$). #### Blending Function $F_1$ $F_1$ is a function of wall distance $d$, $k$, $\omega$, and mean strain rate. - $F_1 = 1$ near the wall $\implies$ pure k-$\omega$ behaviour. - $F_1 = 0$ in the free stream $\implies$ k-$\epsilon$ behaviour. #### SST Eddy Viscosity with Shear Stress Limiter The most important modification of SST is the eddy viscosity limiter: $$\mu_t = \frac{\rho a_1 k}{\max(a_1 \omega, S F_2)}$$ where $S = \sqrt{2 \overline{S}_{ij} \overline{S}_{ij}}$ is the mean strain rate magnitude, and $F_2$ is a second blending function. This limiter enforces the Bradshaw assumption ($\overline{u'v'} = a_1 k$) in adverse pressure gradient regions, preventing overproduction of $\mu_t$ (a key failure of standard k-$\epsilon$ in separating flows). #### Strengths - **Best-in-class** for adverse pressure gradient flows, mild separation, airfoil stall. - **No free-stream sensitivity** (unlike base k-$\omega$). - Industry standard in aerospace and turbomachinery CFD. - Does not require wall functions; integrates directly to the wall. **Relevant Questions:** Q69 ### Direct Numerical Simulation (DNS) #### Definition DNS solves the Navier-Stokes equations exactly on a grid fine enough to resolve all turbulent scales down to the Kolmogorov microscale $\eta$. No turbulence model is used. #### Why DNS Resolves All Scales: Link to Kolmogorov Scale DNS aims to capture turbulent dynamics without any modelling assumption. - Energy is produced at large scales ($L$). - Cascaded through inertial range. - Dissipated at the Kolmogorov scale $\eta = (\nu^3/\epsilon)^{1/4}$. If the grid does not resolve down to $\eta$, energy that should have dissipated remains in the system, leading to numerical pile-up (bottleneck) and erroneous statistics. #### Resolution Requirement - **Grid spacing:** $h \le \eta$ (often $h \sim 2\eta$ in practice). - **Number of grid points per direction:** $N \sim L/\eta$. - Using $\epsilon \sim \overline{U}^3/L$, it implies $N \sim Re^{3/4}$. - **Total grid points:** $N_{total} \sim N^3 \sim Re^{9/4}$. - **Time step:** Limited by Kolmogorov time scale $\tau_\eta = (\nu/\epsilon)^{1/2}$, so $\Delta t \sim \tau_\eta \sim Re^{-1/2}$. - **Total operations:** $\sim Re^{11/4}$. #### Numerical Methods for DNS - **Key Criteria:** 1. **Low numerical dissipation:** Must not contaminate physical $\epsilon$. 2. **Low dispersion error:** Phase relationships among Fourier modes must be preserved. 3. **High order of accuracy:** 4th order or higher; truncation error $\ll$ smallest physical scales. - **Pseudo-spectral methods:** Gold standard for canonical flows due to exponential convergence (Fourier expansion in periodic directions; differentiation exact). - **Compact finite difference:** High-order implicit schemes with low dispersion error. - **Spectral element:** Combines spectral accuracy with geometric flexibility. - **Discontinuous Galerkin:** High-order, locally conservative, well-suited for unstructured grids. #### HPC Advances and Future of DNS - **Trends Enabling Future DNS:** Exascale computing, GPU acceleration, mixed precision, Adaptive Mesh Refinement (AMR), Machine-learning-assisted DNS. - **Remaining Barriers:** Memory bandwidth, I/O for petabytes of data, wall-bounded flows scaling worse ($Re^{37/14}$). - **Outlook:** DNS will increasingly serve as a numerical experiment for generating high-fidelity data to train data-driven turbulence models. #### Limitations for Real Engineering Systems - **Reynolds number ceiling:** Limited to $\sim 10^4-10^5$; industrial flows operate at $10^6-10^9$. - **Computational cost:** $N \sim Re^{9/4}$ scaling makes high-Re DNS infeasible. - **Geometric complexity:** Spectral methods require simple geometries; complex geometries reduce accuracy or feasibility. - **Boundary conditions:** Inlet/outlet treatment for non-periodic flows is non-trivial. - **Multiphysics:** Coupling with combustion, multiphase, FSI, radiation rapidly inflates cost. - **Data deluge:** Storing instantaneous fields generates terabytes per run. - **Turnaround time:** A single DNS may take weeks to months, incompatible with engineering design cycles. - **Wall-bounded scaling:** Worse than $Re^{9/4}$ for resolving inner layer. **Relevant Questions:** Q27, Q28, Q29, Q30, Q31 ### Large Eddy Simulation (LES) #### Definition LES is a turbulence simulation approach that: - **Resolves** the large, energy-containing eddies explicitly (on the computational grid). - **Models** the small-scale (sub-grid scale, SGS) eddies, which are more universal and isotropic, via a SGS model. This is motivated by the energy cascade: large eddies carry $\sim 80\%$ of the energy and are geometry-dependent; small eddies are nearly isotropic and their behaviour is more predictable. #### The Filtering Operation LES applies a spatial filter $G$ of width $\Delta$ to decompose the velocity field: $u_i(\mathbf{x}, t) = \overline{u}_i(\mathbf{x}, t) + u''_i(\mathbf{x}, t)$ where $\overline{u}_i$ is the resolved (filtered) field and $u''_i$ is the residual (sub-grid) field. The filtered field is defined by convolution: $$\overline{u}(\mathbf{x}, t) = \int G(\mathbf{x} - \mathbf{x}'; \Delta) u(\mathbf{x}', t) d\mathbf{x}'$$ Common filter kernels include: - **Box filter:** $G = 1/\Delta$ for $|\mathbf{x}-\mathbf{x}'| ### Sub-Grid Scale (SGS) Modeling in LES #### Context SGS modelling closes the residual stress $\tau^{sgs}_{ij}$ in LES. The most common approach is the eddy-viscosity hypothesis for SGS stresses. #### The SGS Stress Tensor $\tau^{sgs}_{ij} = \overline{u_i u_j} - \overline{u}_i \overline{u}_j$ represents the effect of the unresolved sub-grid scale (SGS) eddies on the resolved flow. It must be modelled because $\overline{u_i u_j}$ (filtered product) is not directly accessible in simulation. #### Smagorinsky Model (1963) The most widely used SGS model applies the Boussinesq hypothesis to the SGS stresses: $$\tau^{sgs}_{ij} - \frac{1}{3} \tau^{sgs}_{kk} \delta_{ij} = -2\nu_{sgs} \overline{S}_{ij}$$ where $\overline{S}_{ij} = \frac{1}{2} \left( \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial \overline{u}_j}{\partial x_i} \right)$ is the resolved strain rate tensor. The SGS eddy viscosity is: $\nu_{sgs} = (C_s \Delta)^2 |\overline{S}|$, where $|\overline{S}| = \sqrt{2\overline{S}_{ij}\overline{S}_{ij}}$. The Smagorinsky constant $C_s \approx 0.1-0.2$. The SGS model introduces an energy drain from the resolved scales to the sub-grid scales, mimicking the forward energy cascade. #### Physical Significance of SGS Viscosity 1. **Energy drain:** Models the forward energy cascade from resolved to unresolved scales. 2. **Numerical stabilization:** Dissipates energy at the grid scale, preventing accumulation and spurious oscillations. 3. **Universality:** Since SGS scales lie in the inertial subrange, SGS models can leverage Kolmogorov universality. #### Limitations of Smagorinsky Model - $C_s$ is not universal; varies between flows and regions. - Cannot represent backscatter (energy transfer from small to large scales). - Overdissipative near walls; requires van Driest damping. #### Dynamic Smagorinsky Model (Germano et al. 1991) Computes $C_s$ locally in space and time using a test-filter. - Requires no user-specified $C_s$. - Automatically goes to zero near solid walls. - Allows negative $C_s$ in averaged sense, representing backscatter. #### Other SGS Models - **Scale-Similarity Model (Bardina et al. 1980):** Assumes SGS stresses are similar to the resolved stress at the smallest resolved scales. Better representation of backscatter but under-dissipative alone. - **Wall-Adapting Local Eddy Viscosity (WALE) Model:** Correctly gives $\nu_{sgs} \sim y^3$ near a wall (no damping function needed). **Relevant Questions:** Q12, Q32, Q41, Q72 ### Kolmogorov Microscales #### Context Kolmogorov's 1941 (K41) theory postulates that the small-scale structure of turbulence is universal – independent of the large-scale geometry and determined entirely by two parameters: the kinematic viscosity $\nu$ and the mean energy dissipation rate $\epsilon$. #### Assumptions - Locally isotropic turbulence at the smallest scales. - High Reynolds number ($Re \gg 1$), so there is a clear separation between energy-containing scales and dissipation scales. - The energy cascade is in statistical equilibrium: energy cascades from large eddies to small eddies at the same rate $\epsilon$ at which it is dissipated. #### Dimensional Analysis (Derivation) **1. Kolmogorov Length Scale ($\eta$):** Seeking $\eta \sim \nu^a \epsilon^b$. Dimensions: $[L] = [L^2 T^{-1}]^a [L^2 T^{-3}]^b$. Equating powers: $1 = 2a + 2b$, $0 = -a - 3b$. Solving yields $a = 3/4, b = -1/4$. $$\eta = \left( \frac{\nu^3}{\epsilon} \right)^{1/4}$$ **2. Kolmogorov Time Scale ($\tau_\eta$):** Seeking $\tau_\eta \sim \nu^a \epsilon^b$. Dimensions: $[T] = [L^2 T^{-1}]^a [L^2 T^{-3}]^b$. Equating powers: $0 = 2a + 2b$, $1 = -a - 3b$. Solving yields $a = 1/2, b = -1/2$. $$\tau_\eta = \left( \frac{\nu}{\epsilon} \right)^{1/2}$$ **3. Kolmogorov Velocity Scale ($u_\eta$):** Seeking $u_\eta \sim \nu^a \epsilon^b$. Dimensions: $[L T^{-1}] = [L^2 T^{-1}]^a [L^2 T^{-3}]^b$. Equating powers: $1 = 2a + 2b$, $-1 = -a - 3b$. Solving yields $a = 1/4, b = 1/4$. $$u_\eta = (\nu \epsilon)^{1/4}$$ #### Verification: Reynolds Number at Kolmogorov Scale The Reynolds number at the Kolmogorov scale is $Re_\eta = \frac{u_\eta \eta}{\nu} = \frac{(\nu \epsilon)^{1/4} (\nu^3/\epsilon)^{1/4}}{\nu} = \frac{\nu}{\nu} = 1$. This confirms that at Kolmogorov scales, viscous forces balance inertial forces – energy is dissipated into heat at these scales. #### Scale Separation with the Integral Scale $L$ For a flow with integral length scale $L$ and velocity scale $\overline{U}$, the energy dissipation rate scales as $\epsilon \sim \overline{U}^3/L$. Substituting this into the expression for $\eta$: $$\frac{\eta}{L} \sim Re^{-3/4}$$ This implies that as the Reynolds number increases, the length scale of dissipative eddies becomes much smaller relative to the integral scale, requiring finer resolution for DNS. #### Importance of Kolmogorov Microscale in CFD The Kolmogorov microscale is the fundamental resolution barrier in turbulence simulation. 1. **DNS Grid Requirement:** To resolve all eddies, the grid spacing $\Delta x \le \eta$. Total grid points scale as $N^3 \sim (L/\eta)^3 \sim Re^{9/4}$. For $Re = 10^6$, $N \sim 10^{13.5}$ points, infeasible. 2. **LES Filter Width:** In LES, $\Delta$ is chosen well above $\eta$ but in the inertial subrange. The Kolmogorov scale dictates how much sub-grid modelling is required. 3. **Time Step Constraint:** The Kolmogorov time scale $\tau_\eta$ dictates temporal resolution. 4. **Universality:** Below $\eta^{-1}$, turbulence statistics are universal (Kolmogorov's hypotheses), enabling generic SGS models. **Relevant Questions:** Q19, Q28, Q39, Q40, Q40 ### Richardson's Energy Cascade #### Concept (L.F. Richardson, 1922) Turbulent kinetic energy is injected at large scales (containing the integral length scale $L$), transferred conservatively through an inertial range to progressively smaller eddies, and finally dissipated at the smallest (Kolmogorov) scales by viscosity. Richardson famously summarized this as: "Big whorls have little whorls that feed on their velocity, and little whorls have lesser whorls and so on to viscosity." #### Diagrammatic Illustration ``` log E(к) ^ | Energy-containing Inertial sub-range Dissipation | range (E ~ к^-5/3) range | . . . | . Energy transfer at rate ε | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . > log к (wave number) +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- к ~ 1/L к ~ 1/η ``` #### Physical Interpretation - **Energy injection:** Occurs at large scales (low $\kappa$) by mean shear or external forcing. - **Inertial sub-range:** Energy is transferred from larger to smaller eddies via vortex stretching, with negligible viscous dissipation. Energy flux $\Pi = \epsilon$ (constant). - **Kolmogorov scale $\eta$:** Viscous forces dominate and dissipate the energy as heat. **Relevant Questions:** Q13 ### Energy Dissipation from Mean and Fluctuating Fields #### Mean Flow Kinetic Energy Equation Define mean kinetic energy $K = \frac{1}{2} \overline{U}_i \overline{U}_i$. Multiplying RANS by $\overline{U}_i$ and averaging: $$\frac{\partial K}{\partial t} + \overline{U}_j \frac{\partial K}{\partial x_j} = -\overline{U}_i \frac{\partial \overline{P}}{\partial x_i} + \overline{U}_i \nu \frac{\partial^2 \overline{U}_i}{\partial x_j \partial x_j} - \overline{U}_i \frac{\partial (\overline{u'_i u'_j})}{\partial x_j}$$ This can be rewritten as: $$\frac{\partial K}{\partial t} + \overline{U}_j \frac{\partial K}{\partial x_j} = P_k - \epsilon_M + \text{transport terms}$$ where: - $P_k = -\overline{u'_i u'_j} \frac{\partial \overline{U}_i}{\partial x_j}$ (production: transfers $K \to k$). - $\epsilon_M = \nu \overline{\frac{\partial \overline{U}_i}{\partial x_j} \frac{\partial \overline{U}_i}{\partial x_j}}$ (mean dissipation). #### Total Energy Budget Adding the mean flow kinetic energy equation and the TKE equation: Total dissipation = $\epsilon_M + \epsilon$. #### Conclusion Energy is dissipated from both the mean field (at rate $\epsilon_M \sim \nu (\partial \overline{U}/\partial y)^2$) and the fluctuating field (at rate $\epsilon \sim \nu (\partial u'/\partial x_j)^2$). #### Order of Magnitude At high $Re$, $\epsilon \gg \epsilon_M$, since fluctuating gradients (at Kolmogorov scale) far exceed mean-flow gradients. However, $\epsilon_M$ is NOT zero; it represents direct viscous dissipation by the mean shear, particularly important in the viscous sublayer of wall-bounded flows. **Relevant Questions:** Q37, Q44, Q45, Q47 ### Isotropy and Homogeneity of Turbulence #### Homogeneity **Definition:** Statistical properties are invariant under spatial translation: $\overline{\phi(\mathbf{x}, t)} = \overline{\phi(\mathbf{x} + \mathbf{r}, t)} \ \forall \mathbf{r}$. - All spatial derivatives of statistical quantities vanish: $\frac{\partial \overline{\phi}}{\partial x_i} = 0$. - **Implication for TKE equation:** For homogeneous turbulence, the divergence term in the TKE equation is zero: $\frac{\partial}{\partial x_j} \left( \frac{1}{2} \overline{u'_i u'_i u'_j} + \frac{1}{\rho} \overline{p' u'_j} - \nu \frac{\partial k}{\partial x_j} \right) = 0$. The TKE equation simplifies to $\frac{Dk}{Dt} = P_k - \epsilon$. This makes Homogeneous Isotropic Turbulence (HIT) a vital canonical flow for fundamental studies (e.g., DNS of decaying turbulence in periodic boxes), since transport is eliminated and one can isolate the production-dissipation balance. #### Isotropy **Definition:** Statistical properties are invariant under rotation and reflection. - $\overline{u'_i u'_j} = \frac{1}{3} \overline{u'_k u'_k} \delta_{ij}$. - No preferred direction; all normal stresses equal; off-diagonal Reynolds stresses vanish. #### Hierarchical Relationship Isotropy implies homogeneity, but not vice versa. Homogeneous Isotropic Turbulence (HIT) is the simplest theoretically tractable form of turbulence and the testbed for Kolmogorov's theory. **Relevant Questions:** Q18, Q38 ### Correlation Length, Time, and Autocorrelation #### Autocorrelation Function **Definition:** The autocorrelation function of a stationary random process $u'(t)$ is: $$R(\tau) = \frac{\overline{u'(t) u'(t + \tau)}}{\overline{u'^2}}$$ or, in non-normalized form: $\rho(\tau) = \overline{u'(t) u'(t + \tau)}$. **Properties:** 1. $R(0) = 1$ (perfect self-correlation at zero lag). 2. $|R(\tau)| \le 1$ (Cauchy-Schwarz). 3. $R(\tau) = R(-\tau)$ (even function). 4. $R(\tau) \to 0$ as $\tau \to \infty$ (loss of memory). #### Key Quantities Derived - **Integral Time Scale ($T_E$):** $T_E = \int_0^\infty R(\tau) d\tau$. Represents the longest correlation time – physically, the lifetime of the largest energy-containing eddies. - **Taylor Microscale ($\lambda_T$):** $\lambda_T = \sqrt{-2/R''(0)}$. - **Power Spectral Density ($S(\omega)$):** $S(\omega) = \frac{1}{2\pi} \int_{-\infty}^\infty R(\tau) e^{-i\omega\tau} d\tau$ (Wiener-Khinchin theorem). #### Integral Length Scale ($L_E$) Using two-point spatial correlation $f(r) = \overline{u'(\mathbf{x}) u'(\mathbf{x}+\mathbf{r})}/\overline{u'^2}$: $L_E = \int_0^\infty f(r) dr$. Physical size of the largest eddies. #### Significance in Turbulent Flows 1. **Determines RANS/LES domain size:** Domain $\gg L_E$ to avoid spurious correlations. 2. **Determines averaging time:** For ergodic estimates, sample time $\gg T_E$. 3. **Inflow conditions:** Synthetic turbulence generators must match $L_E, T_E$. 4. **Connection to Reynolds number:** $Re_L = \overline{U}_{rms} L_E / \nu$. 5. **Taylor's frozen turbulence hypothesis:** $L_E = \overline{U} T_E$, allowing temporal data to be reinterpreted as spatial. **Relevant Questions:** Q23, Q33, Q43, Q52 ### Spectral Filtering via Transfer Function #### Context In LES, filtering in physical space is a convolution. Applying the Fourier transform to this convolution leads to spectral filtering. #### Spectral Space Operation The filtered velocity field $\overline{u}(\kappa)$ in spectral space is related to the unfiltered field $u(\kappa)$ by the transfer function $G(\kappa)$ of the filter: $$\overline{u}(\kappa) = G(\kappa) u(\kappa)$$ #### Common Filter Transfer Functions | Filter | $G(r; \Delta)$ | $G(\kappa)$ | | :------------ | :------------------------------------------- | :-------------------------------------------- | | **Sharp Spectral** | N/A | $1$ for $|\kappa| \le \pi/\Delta$, $0$ else | | **Box (top-hat)** | $1/\Delta$ for $|r| \le \Delta/2$, $0$ else | $\frac{\sin(\kappa \Delta/2)}{\kappa \Delta/2}$ | | **Gaussian** | $\frac{1}{\sqrt{2\pi}\sigma} e^{-r^2/(2\sigma^2)}$ | $e^{-\kappa^2 \Delta^2/24}$ | #### Procedure for Filtering 1. **Take FFT** of velocity signal: $u(\mathbf{x}) \to u(\kappa)$. 2. **Multiply by transfer function:** $\overline{u}(\kappa) = G(\kappa) u(\kappa)$. 3. **Inverse FFT** to recover filtered field: $\overline{u}(\mathbf{x}) = \mathcal{F}^{-1}\{\overline{u}(\kappa)\}$. #### Insight The sharp spectral cutoff filter cleanly separates resolved and SGS scales but is non-local. The Gaussian filter is local in both spaces, making it popular in pseudo-spectral DNS-based *a priori* testing. **Relevant Questions:** Q25, Q26 ### Estimating Wind Tunnel Background Turbulence Intensity #### Definition Turbulence intensity $I$ is defined as: $$I = \frac{U_{rms}}{\overline{U}_\infty} = \frac{\sqrt{\overline{u'^2} + \overline{v'^2} + \overline{w'^2}}}{\overline{U}_\infty}$$ For a single wire, assuming isotropy, $U_{rms} = \sqrt{\overline{u'^2}}$. #### Step-by-Step Procedure 1. **Calibration of Hot-Wire:** Calibrate the hot-wire anemometer in a low-turbulence region against a known reference (e.g., Pitot tube). Fit King's law: $E^2 = A + B\overline{U}^n$. 2. **Probe Placement:** Mount the single-wire probe at the centreline of the test section, oriented perpendicular to mean flow. 3. **Set Tunnel Speed:** Run the wind tunnel at a target velocity $\overline{U}_\infty$. 4. **Data Acquisition:** Record voltage signal $E(t)$ at high sampling rate ($f_s > 10 \text{ kHz}$) for sufficient duration ($T \ge 30 \text{ s}$). Use anti-aliasing filter at $f_s/2$. 5. **Convert Voltage to Velocity:** Apply the calibration: $U(t) = \left( \frac{E^2(t) - A}{B} \right)^{1/n}$. 6. **Reynolds Decomposition:** Compute mean and fluctuation: $\overline{U} = \frac{1}{T} \int_0^T U(t) dt$, $u'(t) = U(t) - \overline{U}$. 7. **RMS Computation:** $U_{rms} = \sqrt{\frac{1}{T} \int_0^T u'^2(t) dt}$. 8. **Compute Intensity:** $I = \frac{U_{rms}}{\overline{U}} \times 100\%$. 9. **Verify with Spectrum:** Compute the power spectral density (PSD) using FFT to ensure no spurious peaks. #### Practical Note Good wind tunnels achieve $I ### Wall Jet Linear Spreading via Momentum Integral #### Setup & Assumptions A 2D wall jet flows along a wall in the x-direction with virtual origin at $x=0$. Assume: - Steady, incompressible, 2D. - Boundary layer approximation: $\partial/\partial x \ll \partial/\partial y$. - Negligible pressure gradient. - Self-similar profile: $\overline{U}/\overline{U}_m = f(\eta)$, $\eta = y/\delta(x)$. #### Momentum Integral Equation The boundary-layer momentum equation: $$\overline{U} \frac{\partial \overline{U}}{\partial x} + \overline{V} \frac{\partial \overline{U}}{\partial y} = \nu \frac{\partial^2 \overline{U}}{\partial y^2} - \frac{\partial (\overline{u'v'})}{\partial y}$$ Integrating from $y=0$ to $\infty$ (using continuity) and assuming negligible $\tau_w$ at large $x$: $$M_0 = \int_0^\infty \overline{U}^2 dy = \text{const}$$ #### Detailed Argument for Linear Spread Assume $\delta(x) \propto x^a$ and $\overline{U}_m(x) \propto x^b$. From momentum conservation, $2b+a=0$. From entrainment hypothesis (entrainment velocity $v_e \sim \overline{U}_m$ at the jet edge and $v_e \sim \frac{d\delta}{dx} \overline{U}_m$), self-similarity dictates $a=1, b=-1/2$. #### Result - $\delta(x) \propto x$ (linear spread) - $\overline{U}_m(x) \propto x^{-1/2}$ (centreline decay) #### Physical Interpretation The wall jet's linear lateral spread is the signature of self-similar turbulent free-shear-layer-like behaviour. Although the wall constrains one side, the outer free-shear layer dominates the spreading. **Relevant Questions:** Q22, Q51 ### Stationary vs Non-Stationary Processes #### Definition: Stationary Process A random process $\phi(t)$ is **statistically stationary** if all its statistical moments are invariant under time translation: $\overline{\phi(t_1) \phi(t_2) \cdots \phi(t_n)} = \overline{\phi(t_1 + \tau) \phi(t_2 + \tau) \cdots \phi(t_n + \tau)}$ #### Weak Stationarity Requires only the mean and autocorrelation to be time-invariant. #### Property Comparison | Property | Stationary | Non-Stationary | | :------------- | :---------------------------------------- | :-------------------------------------------- | | **Mean** | Constant in time | Varies with time | | **Variance** | Constant | Varies with time | | **Autocorrelation** | Depends only on lag $\tau$ | Depends on both $t$ and $\tau$ | | **Examples** | Fully developed pipe flow, steady boundary layer, decaying HIT in moving frame | Atmospheric turbulence (diurnal cycle), decaying turbulence behind a grid, turbulent flow in IC engine cylinder, pulsatile arterial flow | #### Importance Stationarity allows replacement of ensemble averages with time averages (ergodic hypothesis), which is essential for experimental measurements. Non-stationary flows require ensemble averaging over multiple realizations or phase-averaging. **Relevant Questions:** Q23 ### Reynolds Rules of Averaging #### Context These rules are fundamental to the Reynolds decomposition and the derivation of RANS equations. #### Rules For two random variables $f, g$ and a constant $c$: - **Linearity:** $\overline{f+g} = \overline{f} + \overline{g}$ - **Constant Factor:** $\overline{cf} = c\overline{f}$ - **Mean of a Mean:** $\overline{\overline{f}} = \overline{f}$ - **Mean of a Fluctuation:** $\overline{f'} = 0$ - **Mean of Product of Mean and Fluctuation:** $\overline{\overline{f}g'} = \overline{f}\overline{g'} = 0$ - **Commutativity with Differentiation:** $\overline{\frac{\partial f}{\partial x_i}} = \frac{\partial \overline{f}}{\partial x_i}$ and $\overline{\frac{\partial f}{\partial t}} = \frac{\partial \overline{f}}{\partial t}$ - **Crucial:** $\overline{fg} \neq \overline{f}\overline{g}$ (generally $\overline{fg} = \overline{f}\overline{g} + \overline{f'g'}$), which is the origin of the Reynolds stress and the closure problem. #### Significance These rules permit averaging operations on the Navier-Stokes equations to produce closed-form RANS equations. The fact that $\overline{fg} \neq \overline{f}\overline{g}$ is the single most important aspect, leading to the Reynolds stress and the closure problem. **Relevant Questions:** Q20 ### Reynolds Shear Stress Profile in Turbulent Boundary Layer #### Sketch and Description ``` y/δ ^ | edge of BL | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | / | / | / Peak (~ 0.9-1.0 u_tau^2) | / \ | / \ |/___________________\__________________________________________________________ > -u'v'/u_tau^2 0 at wall viscous sublayer ``` #### Salient Features - **At the wall ($y=0$):** $-\overline{u'v'} = 0$ since $u' = v' = 0$ (no-slip). - **Viscous sublayer ($y^+ ### Taylor's Experiment on Transition to Turbulence #### Context: Taylor-Couette Flow G.I. Taylor (1923) studied flow between two concentric cylinders (inner rotating at $\Omega_1$, outer stationary). This geometry (Taylor-Couette flow) provides a controlled setting to observe the sequence of transitions from laminar to turbulent flow. #### Taylor Number Taylor defined the dimensionless parameter: $$Ta = \frac{\Omega_1^2 R_1 d^3}{\nu^2}$$ where $d = R_2 - R_1$ (gap width). #### Sequence of Transitions Observed (Increasing $\Omega_1$) | Stage | Flow state | Physical description | | :----------- | :-------------------------- | :------------------------------------------------------------------------------ | | $Ta Ta_{c1}$ | Taylor vortex flow | Centrifugal instability causes Taylor cells to develop. | | $Ta \gg Ta_{c1}$ | Wavy Taylor vortices | Azimuthal waviness develops; time-periodic, 3D. | | Higher $Ta$ | Modulated wavy vortices | Quasi-periodic state; multiple incommensurate frequencies. | | $Ta \to \infty$ | Turbulent Taylor vortices | Chaotic, turbulent; Taylor vortices persist as coherent structures. | #### Rayleigh's Centrifugal Instability Criterion Taylor showed that the flow is unstable to axisymmetric disturbances when $\frac{d}{dr}(r^2 \Omega)^2 R_1$. #### Significance of Taylor's Work - First quantitative experimental confirmation of a stability theory. - Demonstrated that transition can be continuous and ordered (via bifurcations) rather than catastrophic. - The Taylor number at first instability: $Ta_{c1} \approx 1708$. - Laid the foundation for dynamical systems approaches to turbulence transition. **Relevant Questions:** Q35, Q36, Q65 ### Sensitive Dependence on Initial Conditions (SDIC) #### Concept A hallmark of nonlinear dynamical systems is SDIC: infinitesimally small differences in initial states grow exponentially in time, making long-term prediction practically impossible. This underpins the "apparent randomness" of turbulent velocity signals. #### Lyapunov Exponent The divergence rate is characterised by the Lyapunov exponent $\lambda$: $$\delta(t) \sim \delta_0 e^{\lambda t}$$ For $r=3.9$ in the logistic map example, $\lambda \approx +0.45 > 0$, confirming chaos. #### Connection to Navier-Stokes Equations The N-S equations contain the nonlinear advection term $\overline{U}_j \frac{\partial \overline{U}_i}{\partial x_j}$. At high Re, this nonlinearity dominates and the flow exhibits SDIC. - Turbulent flows are deterministic but chaotic: identical boundary conditions yield statistically equivalent, not identical, realisations. - DNS can only reproduce statistically consistent solutions. - Long-time prediction requires statistical (RANS) rather than deterministic methods. **Relevant Questions:** Q5, Q7 ### Orthogonality of Fourier Modes & Energy Spectrum via Auto-correlation #### Context In turbulence analysis, the velocity signal $u(t)$ is decomposed into a superposition of harmonic (Fourier) modes. The mathematical property that makes this decomposition unique and energy-preserving is the mutual orthogonality of these modes. The auto-correlation function then provides a direct statistical bridge to the energy spectrum via the Wiener-Khinchin theorem. #### Orthogonality of Fourier Modes **Definition of Fourier basis functions:** Consider two complex exponential basis functions at angular frequencies $\omega_m$ and $\omega_n$ defined on the interval $[-T/2, T/2]$: $\phi_m(t) = e^{i\omega_m t}$, $\phi_n(t) = e^{i\omega_n t}$, where $\omega_k = \frac{2\pi k}{T}$ for $k \in \mathbb{Z}$. **Inner product / Orthogonality condition:** The inner product of two basis functions over the period $T$ is: $$\langle \phi_m, \phi_n \rangle = \frac{1}{T} \int_{-T/2}^{T/2} \phi_m(t) \overline{\phi_n(t)} dt$$ - **Case 1: $m \neq n$ (distinct modes):** $\langle \phi_m, \phi_n \rangle = 0$. - **Case 2: $m = n$ (same mode):** $\langle \phi_m, \phi_m \rangle = 1$. Combining both cases: $\langle \phi_m, \phi_n \rangle = \delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta. The Fourier modes form a complete orthonormal set. #### Fourier Representation of the Turbulent Signal A stationary turbulent velocity signal $u(t)$ can be expanded as: $$u(t) = \sum_{n=-\infty}^\infty \hat{u}_n e^{i\omega_n t}, \quad \hat{u}_n = \frac{1}{T} \int_{-T/2}^{T/2} u(t) e^{-i\omega_n t} dt$$ #### Energy Spectrum from the Auto-correlation Function **1. Auto-correlation Function (ACF):** For a stationary random process $u(t)$, the auto-correlation function $R(\tau)$ is: $$R(\tau) = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} u(t) u(t + \tau) dt = \overline{u(t) u(t + \tau)}$$ At $\tau=0$, $R(0) = \overline{u^2}$, which is the variance (turbulent kinetic energy for zero-mean fluctuation). **2. Relationship with Fourier Coefficients:** $$R(\tau) = \sum_n |\hat{u}_n|^2 e^{i\omega_n \tau}$$ **3. Power Spectral Density (PSD):** Taking the Fourier transform of the ACF: $$S(\omega) = \int_{-\infty}^\infty R(\tau) e^{-i\omega\tau} d\tau \quad \text{(Wiener-Khinchin Theorem)}$$ Conversely, the ACF is the inverse Fourier transform of the PSD. At $\tau=0$: $R(0) = \overline{u^2} = \int_{-\infty}^\infty S(\omega) d\omega$. This shows that the PSD $S(\omega)$ gives the distribution of turbulent kinetic energy across frequencies $\omega$. **4. Energy in a Frequency Band:** The energy contained in the frequency band $[\omega_1, \omega_2]$ is: $$E[\omega_1, \omega_2] = \int_{\omega_1}^{\omega_2} E(\omega) d\omega$$ where the one-sided energy spectrum $E(\omega) = 2S(\omega)$ for $\omega > 0$. #### Physical Interpretation The auto-correlation $R(\tau)$ measures how much the flow remembers its own past. A rapidly decorrelating flow implies energy spread across many high frequencies (fine-scale turbulence). The Wiener-Khinchin theorem converts this "memory" information into an energy spectrum directly, without requiring the full time history of each Fourier coefficient. **Relevant Questions:** Q53 ### Laminar vs. Turbulent Flow #### Governing Criterion: Reynolds Number ($Re$) The character of a flow is governed by the Reynolds number: $$Re = \frac{\rho U L}{\mu} = \frac{U L}{\nu}$$ which represents the ratio of inertial to viscous forces. #### Property Comparison | Property | Laminar Flow | Turbulent Flow | | ** **Laminar Flow**:** Smooth, orderly fluid motion in layers. | **Turbulent Flow**:** Chaotic, disordered motion with significant mixing. | | **Reynolds Number ($Re$):** Typically $Re 4000$ for pipe flow. | | **Predictability:** Highly predictable, often described by exact solutions. | Highly unpredictable, requiring statistical or numerical methods. | | **Velocity Profile:** Parabolic in pipes, linear in Couette flow. | Flat in the core, steep near walls (logarithmic profile). | | **Energy Dissipation:** Occurs at the molecular level due to viscosity. | Energy cascades from large to small eddies, dissipated at Kolmogorov scales. | | **Heat Transfer:** Primarily by conduction. | Significantly enhanced by convection due to mixing. | | **Mixing:** Poor. | Excellent. | | **Noise:** Quiet. | Noisy. | **Relevant Questions:** Q2 ### Free Shear Flows #### Definition Turbulent flows that develop in the absence of solid boundaries, generated purely by velocity differences within the fluid. #### Canonical Examples | Flow | Description | Spread Rate | | :---------------- | :-------------------------------------------- | :---------------------------------------------- | | **Free jet (round)** | Fluid issuing into stagnant ambient | $\delta \propto x$, $\overline{U}_c \propto x^{-1}$ | | **Free jet (plane)** | 2D version of above | $\delta \propto x$, $\overline{U}_c \propto x^{-1/2}$ | | **Wake (far)** | Velocity defect behind a body | $\delta \propto x^{1/2}$, $\overline{U}_d \propto x^{-1/2}$ (planar) | | **Mixing layer** | Two parallel streams of different velocity | $\delta \propto x$ | #### Common Features - Self-similarity downstream of a development region. - Linear (or nearly linear) growth of the shear layer width. - No wall constraint – eddies can grow large; boundary-layer hierarchy absent. - Production occurs via shear in the mean velocity profile. - Entrainment of ambient fluid is a key feature. #### Importance Free shear flows are paradigms for: (1) self-similarity, (2) entrainment, (3) Kelvin-Helmholtz instability and coherent structures. They are also crucial in mixing devices, jet propulsion, and pollutant dispersion. **Relevant Questions:** Q49, Q50 ### Significance of the PDF of Velocity Fluctuations #### Definition The Probability Density Function (PDF) $f(u')$ of the velocity fluctuation $u'$ is defined such that: $$P(a \le u' \le b) = \int_a^b f(u')du', \quad \int_{-\infty}^\infty f(u')du' = 1$$ #### Statistical Moments from the PDF All statistical information about $u'$ is encoded in its PDF: - **Mean:** $\overline{u'} = \int_{-\infty}^\infty u' f(u') du' = 0$. - **Variance:** $\overline{u'^2} = \int_{-\infty}^\infty u'^2 f(u') du'$. - **Skewness:** $S = \frac{\overline{u'^3}}{(\overline{u'^2})^{3/2}}$. - **Kurtosis (Flatness):** $F = \frac{\overline{u'^4}}{(\overline{u'^2})^2}$. #### Physical Significance in Turbulence | Property | Physical significance in turbulence | | ** **Practical Use**:** Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | | **Practical Use** | Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | | **Practical Use** | Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | | Practical Use | Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | | **Practical Use** | Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | | **Practical Use** | Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | | Practical Use | Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | | Practical Use | Validating RANS/LES results against experimental PDFs. Intermittency factor $\gamma$: the fraction of time the flow is turbulent. Log-normal or stretched-exponential PDFs at fine scales are signatures of the energy cascade and multifractal turbulence models. | | **Relevant Questions:** Q67 ### Prandtl Mixing Length Theory #### Context The Prandtl mixing length model (1925) was the first successful algebraic turbulence closure. It provides a physical analogy between turbulent momentum transfer and kinetic theory of gases (mean free path of molecules). #### Physical Concept Consider a turbulent shear flow with $\overline{U} = \overline{U}(y)$. A fluid parcel at height $y$ is displaced by a vertical fluctuation $v'$ through a characteristic **mixing length $l_m$** before it mixes with the surrounding fluid. The streamwise velocity fluctuation generated by this displacement: $u' \approx -l_m \frac{d\overline{U}}{dy}$. The vertical fluctuation is assumed of the same order: $|v'| \sim |u'|$. #### Derivation of the Reynolds Stress $$-\overline{u'v'} \approx l_m^2 \left| \frac{d\overline{U}}{dy} \right| \frac{d\overline{U}}{dy}$$ Therefore, the turbulent eddy viscosity is: $$\nu_t = l_m^2 \left| \frac{d\overline{U}}{dy} \right|$$ and the Reynolds shear stress: $$-\rho \overline{u'v'} = \rho l_m^2 \left| \frac{d\overline{U}}{dy} \right| \frac{d\overline{U}}{dy}$$ #### Specification of $l_m$ - **Near wall (inner layer):** $l_m = \kappa y$ (von Kármán constant $\kappa = 0.41$). - **Viscous sub-layer:** $l_m = \kappa y (1 - e^{-y^+/A^+})$ (van Driest damping). - **Outer layer:** $l_m = 0.09 \delta$. - **Free shear flows:** $l_m = C_b \delta(x)$ (proportional to local shear layer width). #### Limitations - $l_m$ must be prescribed – no transport equation. - Cannot handle separated flows, recirculation, or history effects. - $\nu_t = 0$ when $d\overline{U}/dy = 0$ (centreline of pipe) – physically incorrect. Despite these, it gives accurate log-law velocity profiles in attached boundary layers and remains pedagogically invaluable. **Relevant Questions:** Q68