A system and method for estimating wall friction in smooth-wall bounded-turbulent flows

Abstract

A system (100) and method are disclosed for estimating wall friction in smooth- wall-bounded turbulent flows using streamwise mean velocity profile data without requiring near-wall velocity measurements The system (100) receives flow related inputs like flow type, fluid properties, velocity profile U(y) through an input unit (104) and processes the data using a friction estimator (108). A flow check module (204) validates the input data and, for airfoil flows, restricts analysis to a streamwise region of 0.2 < x / c < 0.8. A parameter module (208) computes basic parameters including freestream velocity and characteristic length scale, and integral parameters comprising M, displacement thickness, momentum thickness, and shape factor H. A friction engine (216) evaluates flow separation and applies an empirical friction model using stored constants to estimate friction velocity and wall shear stress.

Description

[0001] “A SYSTEM AND METHOD FOR ESTIMATING WALL FRICTION IN SMOOTH- WALL-BOUNDED TURBULENT FLOWS”

[0002] FIELD OF THE INVENTION:

[0003] The present invention relates generally to a system for estimating wall friction and more particularly relates to a system for estimating wall friction in smooth-wall-bounded turbulent flows.

[0004] BACKGROUND OF THE INVENTION:

[0005] In fluid dynamics, different types of flow describe how fluids move and behave under various conditions. These flow types are categorized based on factors such as velocity, pressure, and the nature of the fluid as, laminar flow, turbulent flow, transitional flow, uniform flow, steady flow, etc. The friction is the factor that affects velocity as well as power consumption when fluid flows over a surface or when adjacent layers of fluid move past each other.

[0006] The turbulent flow is one in which the fluid moves chaotically with irregular fluctuations, eddies, and swirls. It occurs at higher velocities. Fluid particles follow irregular paths, leading to mixing and increased momentum transfer and higher Reynolds number. For example, flow of water in rivers or air flow around an airplane wing are turbulent. When fluid flows through pipes, the resistance between the fluid and the inner surface of the pipe causes friction leading to a loss of pressure in the fluid as it moves through the pipe. Wall friction in pipes is a key factor in fluid transport, affecting both the energy required for pumping and the pressure losses within the system. Understanding how friction works through concepts like Reynolds number, Darcy-Weisbach equation, and the pipe's roughness is essential for designing efficient piping systems.

[0007] Further, in smooth-wall-bounded turbulent flows, estimating the wall friction is essential for understanding the shear stress on the surface due to the fluid's motion. Wall friction arising from the interaction between the turbulent flow and the wall is critical in a variety of engineering applications, including pipe flow, boundary layers, flow over surfaces etc.There are several methods for estimating wall friction due to flows including, partition method, relating wall shear stress to strain rate, Mean-Profile-Based methods, indirect method, etc. These methods have one or more drawbacks such as, flow similarity assumption, inaccuracy, etc. Each method has limitation with respect to the type of flow for which it is applicable.

[0008] The Canadian application CA2570456A1 to Masterov Michael and others describes method and system for multi-path ultrasonic flow measurement of partially developed flow profiles. The method employs an ultrasonic flow meter that includes a plurality of transducers configured to form a plurality of measurement paths in a pipe. The model estimates a Reynolds number for the flowing fluid and compare the estimated Reynolds number with a selected range determining the flow velocity of the flowing fluid based on a flow model. However, turbulence is a potential problem for ultrasonic flow meters and the method is not suitable for all types of flows such as liquids with solid particles. Further, the system and method of the said prior art does not estimate friction.

[0009] The Chinese application CN110207937A to Li Junhong and others describes a method and a system for determining turbulence of an aircraft in consideration of roughness effect. The system utilizes Baldwin-Lomax (BL) algebraic turbulence model that is simulated with the aerodynamic force of a rough surface formed by ablation. However, the Baldwin-Lomax (BL) algebraic turbulence model has several drawbacks such as, the constants in the model are deduced for constant pressure boundary layers at transonic speeds, which may not be suitable for supersonic or hypersonic complex flows, poor performs in case of a large p\rhop is the fluid density. Further the said prior art does not focus on the estimation of friction in a turbulent flow.

[0010] Further, the study titled “A grid and flow adaptive wall-function method for RANS turbulence modelling” to Knopp Tobias, et al. investigates the near-wall RANS solutions of the Spalart-Allmaras and SST k-omega turbulence model near stagnation points and in regions of adverse pressure gradient before separation. These are compared with the corresponding turbulence model specific universal wall-functions. However, a grid and flow adaptive wall-function method can havesome drawbacks, including local mis-predictions in critical flow situations, such as around airfoils and rotor blade. The method has numerical stiffness problems due to large velocity gradients at the wall. Yet again, this study also does not address the problem of estimating friction in turbulent flows.

[0011] There is a need of system and method for estimating wall friction in smoothwall-bounded turbulent flows without requiring near-wall velocity measurements.

[0012] SUMMARY OF THE INVENTION:

[0013] The present invention relates to a system and method for estimating wall friction in smooth-wall-bounded turbulent flows, and more particularly to a computational framework that enables reliable estimation of wall friction parameters without requiring near-wall velocity measurements.

[0014] In accordance with the invention, the system receives flow-related input data, including a streamwise mean velocity profile, flow type information, and fluid properties such as kinematic viscosity. The received inputs are processed by an input module and supplied to a friction estimator that operates through a coordinated sequence of functional modules.

[0015] A flow check module validates the suitability of the received data for friction estimation by identifying the type of flow and, in the case of airfoil flows, verifying that the velocity profile corresponds to a predefined streamwise region where friction estimation is physically meaningful. Invalid or unsuitable input data are flagged to prevent erroneous estimation.

[0016] Upon successful validation, a parameter module computes required flow parameters from the velocity profile. The parameter module includes a basic calculation module for determining fundamental flow quantities such as freestream or centreline velocity and a characteristic length scale, and an advanced calculation module for computing integral flow parameters including displacement thickness, momentum thickness, a kinetic-energy-based parameter, and a shape factor characterizing the turbulent flow structure.

[0017] A friction engine processes the computed parameters to estimate wall friction. The friction engine includes a flow separation check module configured toidentify flows approaching separation based on the shape factor and to exclude such flows from friction estimation. For validated flows not approaching separation, a friction compute module applies a semi-empirical friction model using the computed integral parameters and stored model constants to determine friction velocity and wall shear stress.

[0018] The system further includes a data module comprising a database storing reference parameters and model constants, and a dynamic data module storing user inputs, intermediate computed values, and final estimation results to enable traceability, repeatability, and post-processing analysis. A report module generates output information including estimated friction parameters, flow validation status, and diagnostic messages, which are presented to the user through one or more output devices.

[0019] The invention also provides a method for estimating wall friction that implements the above system architecture and operational workflow. The method includes receiving flow-related inputs, validating flow suitability, computing flow parameters from the velocity profile, excluding flows approaching separation, estimating wall friction using a semi-empirical model, and outputting the estimated friction parameters. Advantageously, the disclosed system and method enable accurate and physically meaningful estimation of wall friction in turbulent flows using outer-layer velocity information, thereby overcoming limitations associated with near-wall measurements and making the invention suitable for experimental, numerical, and practical engineering applications involving limited near-wall data availability.

[0020] BRIEF DESCRIPTION OF DRAWINGS:

[0021] The objectives and advantages of the present invention will become apparent from the following description read in accordance with the accompanying drawings wherein,

[0022] FIG. 1 shows a perspective of a system for estimation of wall friction in smoothwall-bounded turbulent flows in accordance with the present invention;

[0023] FIG. 2 shows a schematic of the system of FIG. 1 ;FIG. 3 shows detailed schematic of the friction estimator of the system of present invention as shown in FIG. 1 ;

[0024] FIG. 3A shows a functional block diagram of the friction estimator of system of FIG. 1;

[0025] FIG. 3B, is a flow diagram illustrating the workflow of the friction estimator of the system of FIG. 1;

[0026] FIG. 4 shows a graphical representation of the profile of streamwise mean velocity U and an example of tabulated values describing the mean velocity profile;

[0027] FIG. 5 shows traditional friction plot for different type of flows;

[0028] FIG. 6 shows the friction values as estimated by the system and method in accordance with the present invention;

[0029] FIG. 7 shows percentage deviation of the actual friction velocity with respect to the value estimated by the system and method in accordance with the present invention; FIG. 8 shows the percentage of data points falling in different deviation bands for friction estimated by the system and method in accordance with the present invention;

[0030] FIG.S 9A-B demonstrate the degradation results for M and H for a cutoff of y+ =40; FIG.S 10A-B demonstrate the degradation results for M and H for a cutoff of y+=80; and

[0031] FIG. 11 illustrates a method flowchart describing sequential steps executed by the system for estimating wall friction of system of FIG. 1.

[0032] DESCRIPTION OF THE INVENTION:

[0033] References in the specification to "one embodiment" or "an embodiment" means that a particular feature, structure, characteristic, or function described in connection with the embodiment is included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment.

[0034] References in the specification to “preferred embodiment” means that a particular feature, structure, characteristic, or function described in detail therebyomitting known constructions and functions for clear description of the present invention.

[0035] The foregoing description of specific embodiments of the present invention has been presented for purposes of illustration and description. They are not intended to be exhaustive or to limit the present invention to the precise forms disclosed and obviously many modifications and variations are possible in light of the above teaching.

[0036] In general aspect, the present invention relates to a system for estimation of wall friction in smooth-wall bounded- turbulent flows.

[0037] FIG. 1 illustrates a system (100) for estimating wall friction in smooth-wall-bounded turbulent flows in accordance with the present invention. The system 100 is configured to receive flow-related inputs, process velocity profile data, and provide estimated wall friction parameters as outputs.

[0038] The system 100 includes a housing (101) enclosing electronic and computational components of the system (100). The housing (101) may be configured as a portable, tablet-like enclosure or as a fixed installation suitable for laboratory or industrial environments. The system (100) further includes an input interface (102) and a display unit (103) positioned on a front surface of the housing 101.

[0039] The input interface (102) enables user interaction with the system (100) and includes one or more input controls, such as physical buttons, touch- sensitive regions, or selectable icons. In the illustrated embodiment, the input interface 102 includes a confirmation control (102a), navigation controls (102b, 102c), and a data input control (104). The input interface (102) allows a user to input or select flow-related parameters including flow type, fluid type, velocity profile data, and fluid properties. In alternative embodiments, the input interface (102) may further include data ports for receiving velocity profile data from external measurement instruments, numerical simulation platforms, or data storage devices.

[0040] The display unit (103) is configured to present information to a user and may include a visual display such as a liquid crystal display or touch screen. The display unit (103) displays input parameters, processing status, and output resultsgenerated by the system (100), including estimated friction velocity and wall shear stress.

[0041] Internally, the system (100) includes a processing unit, memory, and control circuitry (not shown in FIG. 1) configured to execute a friction estimation program stored in the memory. During operation, flow-related input data provided through the input interface (102) is processed by internal computational modules of the system, and the resulting wall friction estimates are presented on the display unit (103).

[0042] The system (100) may operate as a standalone unit or may be communicatively coupled with external experimental setups, numerical simulation platforms, or industrial flow analysis systems. The configuration illustrated in FIG.

[0043] 1 enables practical deployment of the present invention for estimating wall friction without requiring near-wall velocity measurements.

[0044] Referring to FIG. 2, a system (100) for estimation of wall friction in smoothwall-bounded turbulent flows, hereinafter, referred to as system (100) is described. The system (100) includes an input unit (104), a friction estimator (108), and an output unit (112). The system 100 also includes a memory device (116), a controller (120), a control module (124), a database (128), an input module (132) and output module (136). The system (100) receives inputs such as flow type, fluid type, fluid properties, velocity profile, and user identifiers through the input unit (104) wherein the inputs are received via input devices such as a microphone, camera or in the form of graphical diagrams, alpha-numeric inputs and the like. These inputs are received by the input module (132) and processed so as to send those inputs to the friction estimator (108). The output module (136) functions in coordination with the control module (124).

[0045] The input module (132), which is connected to input unit (104), is configured to receive flow-related inputs including flow type, fluid type, streamwise velocity profile, fluid kinematic viscosity, and fluid density. These inputs may be obtained from experimental measurements, numerical simulations, or external flow databases and are initially stored within the input module (132).The collected input data is subsequently transferred to the friction estimator (108) for processing.

[0046] Referring to FIG. 3, the friction estimator (108) is described. The friction estimator (108) includes a flow check module (204), a parameter module (208), a report module (212), a friction engine (216) and a data module (220) each performing a distinct technical function in a sequential and coordinated manner. The friction estimator (108) forms the core analytical component of the system (100) and is configured to process flow-related input data to accurately estimate wall friction in smooth-wall-bounded turbulent flows.

[0047] The flow check module (204) is configured to evaluate the suitability and validity of the received flow data for friction estimation. This module verifies the type of flow, such as boundary layer flow, pipe flow, channel flow, or airfoil flow, and ensures that the velocity profile data corresponds to an appropriate streamwise location. In case of airfoil flows, the flow check module ensures that the velocity profile data lies within a predefined streamwise region, specifically 0.2 < x / c < 0.8, where reliable friction estimation is physically meaningful. If the received data does not satisfy the predefined criteria, the flow check module (204) flags the data as unsuitable and communicates the same to the report module (212), thereby preventing inaccurate friction estimation.

[0048] The parameter module (208) is configured to compute flow parameters required for friction estimation and includes a basic calculation module (BCM) and an advanced calculation module (ACM). The basic calculation module determines fundamental flow quantities such as freestream velocity and boundary layer thickness based on the received velocity profile. The advanced calculation module computes integral flow parameters derived from the velocity profile, including kinetic-energy-based parameter M, displacement thickness, momentum thickness, and shape factor H, which collectively characterize the turbulent flow structure independent of near-wall measurements.

[0049] The friction engine (216) is configured to estimate wall friction using the computed integral parameters. The friction engine includes a flow separation check module (217) and a friction compute module (218). The flow separation checkmodule (217) evaluates the computed shape factor H to identify flows subjected to adverse pressure gradients approaching separation. Flows exceeding a predefined threshold value of H are excluded from friction estimation, as friction becomes physically insignificant in such conditions.

[0050] Upon confirmation that the flow is not approaching separation, the friction compute module (218) is activated to estimate wall friction. The friction compute module (218) applies a predefined semi-empirical friction model using the computed integral flow parameters in combination with stored model constants to determine the friction velocity and the corresponding wall shear stress. The friction compute module thereby generates physically meaningful and reliable friction estimates without requiring near-wall velocity measurements, enabling accurate friction estimation even under conditions of limited near-wall data availability.

[0051] The data module (220) is configured to manage, store, and retrieve data required for operation of the friction estimator (108) and for accurate estimation of wall friction. The data module (220) includes a database (221) and a dynamic data module (222), each serving a distinct technical purpose within the system (100).

[0052] The database (221) is configured to store predefined and static reference data required for friction estimation. In accordance with the present invention such data includes, but is not limited to, empirical friction model constants, threshold H values for flow separation assessment, reference parameters for different flow types, and calibration constants used by the friction compute module. The database (221) enables consistent and repeatable application of the friction estimation model across different flow conditions and operating environments.

[0053] The dynamic data module (222) is configured to store and manage timevarying and session-specific data generated during system operation. The dynamic data module (222) maintains records of user-provided inputs, validated velocity profile data, computed basic and advanced flow parameters, intermediate computational results, and final estimated values of friction velocity and wall shear stress. The dynamic data module (222) further facilitates traceability of computations, repeatability of results, and post-processing analysis, includingvalidation, performance assessment, and historical comparison of friction estimation outcomes.

[0054] The report module (212) of the friction estimator (108) is configured to generate output information including estimated friction velocity, wall shear stress, flow validation status, and diagnostic messages. The report module (212) communicates with the output module to present results through suitable output devices and also reports error conditions or data insufficiencies identified during processing.

[0055] FIG. 3 A shows a functional block diagram of the friction estimator (108) of the system (100) in accordance with the present invention. The friction estimator includes a flow check module (204), a parameter module (208) (not shown), a friction engine (216), a data module (220), and a report module (212) operating in a coordinated manner to estimate wall friction in smooth-wall-bounded turbulent flows.

[0056] The flow check module (204) receives flow-related input data from the input module and validates the suitability of the data for friction estimation. Upon successful validation, the parameter module (208) computes basic flow parameters through a basic calculation module and integral flow parameters through an advanced calculation module. The computed parameters are processed by the friction engine (216), which includes a flow separation check module (217) configured to exclude flows approaching separation, and a friction compute module (218) configured to estimate friction velocity and wall shear stress using stored model constants retrieved from the database (221).

[0057] The data module (220), having a database (221) and a dynamic data module (222), stores reference parameters, model constants, intermediate computational results, and final estimated friction values. The report module (212) generates output information including estimated friction parameters and diagnostic messages, which are communicated to the output module for presentation to the user.

[0058] Referring now to FIG. 3B, a flow diagram illustrating the operational workflow of the friction estimator (108) of the system (100) is described. Thefriction estimator operates as a coordinated sequence of functional modules configured to process flow-related input data and estimate wall friction in smoothwall-bounded turbulent flows.

[0059] The process preferably begins with receipt of validated flow input data including a streamwise mean velocity profile, flow type information, and fluid properties (Step S3A02). The received data is processed by the flow check module (204), which evaluates the suitability of the data for friction estimation by verifying the flow type and, in the case of airfoil flows, confirming that the velocity profile corresponds to a predefined streamwise region of 0.2 < x / c < 0.8 (Steps S3AO3-S3A04). Input data failing the validation criteria is flagged as unsuitable and communicated to the report module (212), thereby preventing further processing (Step S3A05).

[0060] Upon successful validation, the parameter module (208) is executed to compute flow parameters required for friction estimation (Step S3A06). The basic calculation module computes fundamental flow quantities including freestream or centerline velocity and characteristic length scale (Step S3A07), while the advanced calculation module computes integral flow parameters derived from the velocity profile, including kinetic-energy-based parameter M, displacement thickness, momentum thickness, and shape factor H (Step S3AO8).

[0061] The computed shape factor H is evaluated by the flow separation check module (217) to identify flows approaching separation (Steps S3A09-S3A10). Flows exceeding a predefined threshold value of H are excluded from friction estimation, and an exclusion status is communicated to the report module (212 (Steps S3A11-S3A12).

[0062] For validated flows not approaching separation, the friction compute module (218) is activated to estimate wall friction by applying a predefined semi-empirical friction model using the computed integral parameters and model constants retrieved from the database (221) (Step S3A13). The resulting friction velocity and wall shear stress values are stored in the dynamic data module (222) along with intermediate computational results (Step S3A14).Finally, the report module (212) generates output information including estimated friction parameters, flow validation or exclusion status, and diagnostic messages, which are presented to the user through the output module (136), thereby completing the friction estimation process (Steps S3A15-S3A16).

[0063] Now referring to FIGS. 1-3A, the operation of the system 100 for estimating wall friction in smooth-wall-bounded turbulent flows is described. Flow-related inputs including flow type, fluid type, streamwise velocity profile, fluid kinematic viscosity, and fluid density are received through the input unit 104 via one or more input devices. The received inputs are provided to the input module 132, which processes and organizes the input data and subsequently supplies the processed inputs to the friction estimator 108 for further analysis.

[0064] The input data is first analyzed by the flow check module 204 of the friction estimator 108, which determines the suitability of the received velocity profile data for friction estimation. The flow check module 204 identifies the type of flow under consideration, such as boundary layer flow, pipe flow, channel flow, or airfoil flow. In the case of airfoil flows, the flow check module 204 verifies that the velocity profile data corresponds to a predefined streamwise region, specifically 0.2 < x / c < 0.8, where x denotes the distance from the leading edge and c denotes the chord length of the airfoil. Velocity profile data outside this region is identified as unsuitable for accurate friction estimation, and in such cases the flow check module 204 communicates an error or warning message to the report module 212, thereby preventing further processing.

[0065] Upon validation of the input data, the velocity profile data is forwarded to the parameter module 208. The basic calculation module (BCM) of the parameter module computes fundamental flow quantities, including freestream velocity and characteristic boundary layer dimensions, based on the received velocity profile U(y). Subsequently, the advanced calculation module (ACM) computes integral flow parameters derived from the velocity profile, including kinetic-energy-based parameter M, displacement thickness, momentum thickness, and shape factor H, which collectively characterize the turbulent flow independent of near-wall velocity measurements.The computed parameters are then transferred to the friction engine 216, which includes a flow separation check module and a friction compute module. The flow separation check module evaluates the computed shape factor H to determine whether the flow is subjected to an adverse pressure gradient and is approaching separation. Flows exhibiting H > 2, for which friction values approach zero and friction estimation is physically insignificant, are excluded from further processing. In such cases, the flow separation check module directly communicates the exclusion status to the report module 212.

[0066] When the flow is confirmed to be free from separation effects, the friction compute module is activated to estimate wall friction. The friction compute module applies a predefined semi-empirical friction model using the computed integral flow parameters in combination with stored model constants to calculate friction velocity and the corresponding wall shear stress, thereby enabling accurate friction estimation without requiring near-wall velocity data.

[0067] The data module 220 supports the operation of the friction estimator 108 and includes a database and a dynamic data module. The database stores predefined friction model constants and reference parameters, including constants Al, BO, Bl, p, q, and Gref, required for consistent application of the friction estimation model. The dynamic data module stores user inputs, validated velocity profiles, computed basic and advanced parameters, intermediate computational results, and final predicted values of friction velocity and wall shear stress, thereby enabling traceability, repeatability, and post-processing analysis.

[0068] The report module 212 aggregates data and status information received from the input module, flow check module, parameter module, friction engine, and data module, and generates output reports comprising estimated friction values, processing status, and diagnostic messages. These outputs are communicated to the output module 136.

[0069] In the final step, the output module 136 presents the estimated friction velocity, wall shear stress, and related processed data through one or more output devices, including visual displays, data files, or external system interfaces.The control module 124, in communication with the controller 120, coordinates and sequentially executes the operations of all modules of the system 100, thereby ensuring synchronized data flow, controlled execution, and reliable estimation of wall friction.

[0070] Example 1: In one exemplary implementation of the present invention, the system 100 is used to estimate wall friction for a smooth-wall turbulent boundary layer flow over an aerodynamic surface i.e. an airfoil. A streamwise mean velocity profile U(y) corresponding to a turbulent boundary layer is obtained at a streamwise location within the range 0.2 < x / c < 0.8. The velocity profile data, along with associated fluid properties including kinematic viscosity v and fluid density p, is provided to the system 100 through the input unit 104. The received inputs are processed by the input module 132 and supplied to the friction estimator 108.

[0071] The flow check module 204 identifies the flow as an airfoil boundary layer flow and verifies that the velocity profile corresponds to the predefined streamwise region suitable for friction estimation. Upon successful validation, the velocity profile data is forwarded to the parameter module 208.

[0072] The basic calculation module (BCM) computes fundamental flow quantities including the freestream velocity Uco and boundary layer thickness 5 from the received velocity profile U(y). Subsequently, the advanced calculation module (ACM) computes integral flow parameters including the kinetic-energy-based parameter M, displacement thickness 5*, momentum thickness 9, and shape factor H.

[0073] The computed shape factor H is then evaluated by the flow separation check module 217 of the friction engine 216. In this example, the value of H is found to be less than 2, indicating that the flow is not approaching separation and is suitable for friction estimation.

[0074] Upon confirmation that the flow is not approaching separation, the friction compute module 218 applies the predefined empirical friction model of the present invention using the computed integral parameters and stored model constants Ai = 2.0699, Bo = -0.4749, Bi = -0.2860, p = 0.5392, q = -0.5451, and Gref = 6.8. Thefriction compute module 218 calculates the friction velocity Ur, from which the corresponding wall shear stress tw is determined.

[0075] All intermediate parameters and computed friction values are stored in the dynamic data module of the data module 220, while the predefined constants are retrieved from the associated database 221. The report module 212 aggregates the processing status and final estimated values and communicates the results to the output module 136.

[0076] The output module 136 displays the estimated friction velocity and wall shear stress, thereby enabling accurate friction estimation without requiring nearwall velocity measurements.

[0077] Example 2: In another aspect the present invention relates to a method for estimation of wall friction in smooth-wall-bounded turbulent pipe flows. Accordingly, in another exemplary implementation of the present invention, the system 100 is used to estimate wall friction in a smooth-wall turbulent pipe flow.

[0078] A streamwise mean velocity profile U(y) measured across the pipe radius is obtained from either experimental measurements or numerical simulations. The velocity profde data, along with fluid properties including kinematic viscosity v and fluid density p, is provided to the system 100 through the input unit 104. The received inputs are processed by the input module 132 and supplied to the friction estimator 108.

[0079] The flow check module 204 identifies the flow type as pipe flow and verifies that the velocity profile extends from the pipe wall to the centerline, where the centerline velocity represents the freestream velocity Uco. Upon validation of the input data, the velocity profile is forwarded to the parameter module 208.

[0080] The basic calculation module (BCM) computes the centerline velocity and pipe radius, which represents the characteristic length scale 5. The advanced calculation module (ACM) then computes the integral flow parameters, including the kinetic-energy-based parameter M, displacement thickness, momentum thickness, and shape factor H, from the velocity profile U(y).

[0081] The flow separation check module 217 of the friction engine 216 evaluates the shape factor H and confirms that the flow is not subjected to adverse pressuregradients or separation. Upon confirmation, the friction compute module 218 applies the predefined semi-empirical friction model using the computed integral parameters and stored model constants of the present invention to estimate the friction velocity Ur and corresponding wall shear stress TW.

[0082] The computed results are stored in the dynamic data module 222 of the data module 220, and the estimated friction values are reported through the report module 212 to the output module 136, enabling accurate estimation of wall friction without requiring near-wall velocity measurements.

[0083] Example 3: In yet another exemplary implementation, the system 100 is employed to estimate wall friction in a compressible turbulent boundary layer flow, such as a supersonic or hypersonic flow over a smooth aerodynamic surface. A streamwise velocity profile corresponding to the compressible flow is obtained at a suitable streamwise location and provided to the system 100 through the input unit 104, along with fluid properties including kinematic viscosity and density. The received velocity profile is first transformed into an equivalent incompressible velocity profile using a Van Driest transformation, thereby compensating for density variations inherent in compressible flows. The transformed velocity profile is processed by the input module 132 and supplied to the friction estimator 108.

[0084] The flow check module 204 validates the flow type and confirms the suitability of the transformed velocity profile for friction estimation. The validated profile is forwarded to the parameter module 208, wherein the basic calculation module (BCM) computes freestream velocity and boundary layer thickness. The advanced calculation module (ACM) subsequently computes integral parameters including M, displacement thickness, momentum thickness, and shape factor H.

[0085] The flow separation check module 217 evaluates the shape factor H to ensure that the compressible flow is not approaching separation. Upon confirmation, the friction compute module 218 applies the empirical friction model of the present invention using the computed integral parameters and stored constants to estimate friction velocity and wall shear stress.The estimated friction values and associated parameters are stored in the dynamic data module 222 and communicated through the report module 212 to the output module 136. The system thereby enables accurate estimation of wall friction in compressible turbulent flows without requiring high-resolution near-wall measurements.

[0086] Referring to FIG. 4, the streamwise mean velocity U is defined as a function of the wall-normal distance y measured from the surface or wall. The streamwise mean velocity U varies with height y above the wall and satisfies the no-slip boundary condition, wherein U = 0 at the wall (y = 0). As the distance from the wall increases, the velocity increases monotonically in the wall-normal direction.

[0087] In boundary layer flows, the streamwise mean velocity U approaches 0.99Uco at the boundary layer thickness 5, whereas in pipe or channel flows, the velocity reaches the freestream or centreline velocity Uco at the centreline of the pipe or channel, respectively. The velocity profile U(y) thus represents a physically measurable flow characteristic that captures the macroscopic structure of the turbulent flow independent of near-wall resolution. Based on the above-described physical characteristics of the streamwise mean velocity profile, the method of the present invention for estimating wall friction in smooth-wall-bounded turbulent flows is described in forgoing description.

[0088] Accordingly, a person skilled in the art will appreciate that the streamwise mean velocity U is a function of height y from the wall. The streamwise mean velocity U varies with height y above the surface or the wall. Further, U=0 at the wall due to the no- slip condition and U= 0.99 U , at the boundary layer thickness <5 or U= Ua at the centerline of pipe or channel flow.

[0089] The present invention further relates to a method for estimating wall friction in smooth-wall bounded turbulent flows. The method is implementable using the system described with reference to FIGS. 1-3 and is configured to estimate wall friction based on streamwise mean velocity profile data without requiring near-wall velocity measurements. In accordance with the present invention said method is applicable to a wide range of turbulent flows, including boundary layer flows, pipeflows, channel flows, and compressible flows, and enables accurate friction estimation even under conditions of coarse near-wall data resolution.

[0090] The method includes a sequence of above-mentioned steps involving validation of flow data, computation of basic and integral flow parameters, assessment of flow separation, and estimation of friction velocity and wall shear stress using a predefined empirical friction model. The method relies on integral parameters derived from the velocity profile and stored model constants to achieve reliable friction estimation across different flow regimes.

[0091] The notations used for various basic quantities discussed in the method of the present invention is as follows:

[0092] 6 = boundary layer thickness / pipe radius / channel half-height (m)

[0093] U = streamwise (in the direction of the flow) mean velocity (m / s)

[0094] C / oo = freestream velocity outside the boundary layer / centerline velocity of pipe / channel flow (m / s)

[0095] y = distance from the surface or the wall in the direction perpendicular to the wall (m)

[0096] v = kinematic viscosity of the fluid (m2 / s)

[0097] TW= wall shear stress i.e. shear force experienced by the flow at the wall per unit area of the wall (N / m2)

[0098] p = density of the fluid (kg / m3)

[0099] UT= friction velocity (m / s)

[0100] FC = Friction coefficient in the traditional framework defined as 2*( UT / UXY2

[0101] Re = Reynolds number in the traditional framework defined as 6U , / v

[0102] The first step of the method of the present invention includes collecting the inputs for estimation of wall friction in smooth-wall-bounded turbulent flows. Theinputs include (i) the profile U(y) of the streamwise mean velocity U at the location where the friction is to be estimated in a given flow; and (ii) the kinematic viscosity v of the fluid.

[0103] The second step of the method of the present invention includes evaluating the type of flow whose friction is to be obtained. If the flow is over an airfoil, the velocity profile data belonging to the streamwise location in the range of 0.2 < x / c < 0.8 is to be considered for accurate results. Here x is the distance from the leading edge of the airfoil along the chord line and c is the chord length.

[0104] The third step of the method of the present invention includes computing the basic parameters namely the freestream velocity Ux> and the boundary layer thickness 6. For pipe and channel flows, Ux> is the centreline velocity, and 6 is the pipe radius / channel half height.

[0105] For compressible flows, such as supersonic or hypersonic flow, the density variations are significant and the input data is compensated in the form of Van Driest transformation that accounts for these strong variations of density. The Van Driest transformation converts the actual compressible velocity profile to an equivalent incompressible one. Therefore, the input profile U(y) in case of compressible flows is the Van Driest transformed velocity profile.

[0106] The fourth step of the method of the present invention includes computing the integral parameters namely M (which is proportional to the mean streamwise kinetic energy content of the flow), displacement thickness <5*, momentum thickness 0 and shape factor H.

[0107]

[0108] The fifth step of the method of the present invention includes evaluating the flow separation wherein the shape factor H is assessed for an adverse pressure gradient (APG). The value H > 2 indicates that the flow is approaching separation with vanishingly small values of friction. These flows are discarded since the method of the present invention is for computing friction and friction is not relevant in such flows.

[0109] The sixth step of the method of the present invention includes obtaining an estimate value of Ur by employing the equation given below.

[0110]

[0111] , 0.2860 and are known model constants. Further F and R are “effective” dimensionless friction and Reynolds number defined respectively as given below:

[0112]

[0113] , 6.8 that are known constants.

[0114] Accordingly, the method of the present invention proceeds through the following steps for estimating wall friction in smooth-wall-bounded turbulent flows.

[0115] Step 1: Acquisition of Flow Input Data. The method begins with acquiring input data required for estimation of wall friction. The input data includes:

[0116] (i) a streamwise mean velocity profile U(y) corresponding to a location in the flow where wall friction is to be estimated; and

[0117] (ii) the kinematic viscosity v of the fluid.The velocity profile U(y) represents the variation of streamwise mean velocity with respect to the wall-normal distance y from the surface.

[0118] Step 2: Identification and Validation of Flow Type: In the second step, the type of turbulent flow for which friction is to be estimated is identified. The method distinguishes between different flow configurations including boundary layer flow, pipe flow, channel flow, and airfoil flow.

[0119] When the flow corresponds to an airfoil flow, the method includes validating the streamwise location at which the velocity profile is obtained. For accurate friction estimation, the velocity profile data is selected from a predefined streamwise region 0.2 < x / c < 0.8, where x is the distance measured from the leading edge of the airfoil along the chord line and c is the chord length. Velocity profile data obtained outside this region is excluded from further processing, as such data may not yield physically reliable friction estimates.

[0120] Step 3: Computation of Basic Flow Parameters. Upon validation of the input data, the method proceeds to compute basic flow parameters from the velocity profde U(y). These parameters include the freestream velocity Uco and a characteristic flow length scale 5. For boundary layer flows, Uco corresponds to the freestream velocity outside the boundary layer and 5 corresponds to the boundary layer thickness. For pipe and channel flows, Uco corresponds to the centerline velocity, while 5 corresponds to the pipe radius or channel half-height, respectively. In the case of compressible turbulent flows, such as supersonic or hypersonic flows, density variations significantly influence the velocity profile.

[0121] Accordingly, the method includes transforming the measured or computed compressible velocity profile into an equivalent incompressible velocity profile using a Van Driest transformation, which compensates for density variations. The transformed velocity profile is then used for subsequent computations.

[0122] Step 4: Computation of Integral Flow Parameters. Following computation of the basic parameters, the method computes integral flow parameters derived from the velocity profile U(y). These parameters include: a kinetic-energy-based parameter M, which is proportional to the mean streamwise kinetic energy contentof the flow, displacement thickness 5*, momentum thickness 9, and shape factor H, defined as the ratio of displacement thickness to momentum thickness.

[0123] It is noted that these integral parameters provide a global characterization of the turbulent flow structure and are computed by integrating the velocity profile in the wall-normal direction. Importantly, the computation of these parameters is insensitive to near-wall velocity measurements at high Reynolds numbers.

[0124] Step 5: Assessment of Flow Separation. This step includes evaluation whether the flow is subjected to an adverse pressure gradient and is approaching separation. This evaluation is performed by assessing the computed shape factor H. A value of H > 2 indicates that the flow is nearing separation, in which case wall friction values approach zero and friction estimation is no longer physically meaningful. Such flows are therefore excluded from further processing. When H < 2, the flow is identified as suitable for friction estimation and the method proceeds to the next step.

[0125] Step 6: Estimation of Friction Velocity and Wall Shear Stress. Upon confirmation that the flow is not approaching separation, the method estimates the friction velocity Ur by applying a predefined semi-empirical friction model of the present invention. The friction velocity is computed using the integral flow parameters and stored model constants Ai, Bo, Bi, p, q, and Gref.

[0126] Based on the estimated friction velocity Ur, the corresponding wall shear stress rw is determined using known physical relationships. The estimated friction values represent physically meaningful wall friction characteristics of the turbulent flow without requiring near-wall velocity data, especially in high Reynolds number flows as demonstrated later.

[0127] Step 7: Output and Storage of Results. The estimated friction velocity, wall shear stress, and associated computed parameters are stored for record-keeping and post-processing analysis. The results may be displayed, transmitted, or further processed depending on the application requirements.

[0128] EXAMPLES:Only a few examples and implementations are disclosed. Variations, modifications, and enhancements to the described examples and implementations and other implementations can be made based on what is disclosed.

[0129] Examples are set forth herein below and are illustrative of conditions that can be utilized in practicing the disclosure. It will be apparent, however, that the disclosure can be practiced with other conditions than those used in the examples, and the resulting devices various different properties and uses in accordance with the disclosure above and as pointed out hereinafter.

[0130] EXAMPLE 1: Utility Studies

[0131] The method of the present invention provides accurate estimates of friction in a wide variety of flows. 192 experimental / simulation mean velocity profiles were selected from 44 data sets on different kinds of flows.

[0132] As shown in FIG. 6 all the data collapse and tightly cluster around the friction model equation of the present invention. This is in contrast to the complete lack of collapse in the traditional friction plot shown in FIG.5.

[0133] Further FIG. 7 shows the percentage deviations of the friction velocity values of the data points of FIG.6 from those estimated by the model equation shown by the purple line. FIG. 8 shows the histogram of the percentage of data points falling in different deviation bands with respect to the model estimations. It is clear that the percentage deviations are very well limited to the band of ±5% which is typical of the measurement uncertainty given the vast variety of flows being reconciled in the present framework.

[0134] The results as indicated in the FIGS. 5-8 unequivocally confirm the immense utility of the system and method of the present invention in accurately estimating friction in a very wide variety of flows.

[0135] EXAMPLE 2: Robustness Studies

[0136] In practical applications, Reynolds numbers are quite high either due to the large length scale or due to the large velocity of the flow or both. The order of magnitude of the Reynolds number based on friction velocity and boundary layerthickness is typically Rez = 0(10^7) for flows over the aircraft wings and fuselage, atmospheric boundary layers etc. Experimental measurements and numerical simulations of these high-Re flows are extremely challenging in terms of resolving the near-wall flow field. As an example, the boundary layer on the fuselage of a widebody jet airliner could have a thickness 6 = 0.1 m. Assuming a conservative friction Reynolds number Rez =10^6, the distance of y+= y Uz / v = 100 corresponds to y / <5 = 100 / 10A6 = 0.0001 which, with 8 = 0.1 m, translates to y = 0.00001 m = 0.01 mm. Experimentally, it is very challenging to measure at such short distances from the wall and often such measurements are simply outside the capabilities of the measurement systems. For computations as well, these distances are quite challenging because the grid needs to be refined close to the wall to capture the strong variations of mean velocity correctly and this increases the computational costs enormously for real life simulations.

[0137] As such, practical computations and experiments do not have well-resolved near-wall mean velocity profiles which make the estimation of friction very hard in such situations. The problem worsens when one deals with flows subjected to pressure gradients where standard methods of estimating friction are known to fail by a large margin. The system and method of the present invention offers a particularly attractive resolution to these difficulties when one is interested in obtaining friction with coarse resolution of the near-wall data points.

[0138] For the purpose of demonstration of the robustness of the method with reference to the unavailability of near-wall data points due to coarse resolution in the wall-normal direction, it is noted that the method of the present invention involves parameters M and H which are to be obtained by integration of the velocity profile U(y) in the wall-normal direction. Unavailability of near-wall data would therefore degrade these two parameters. To estimate the extent of this degradation, we need a reference for each profile from where the degradation can be measured. A reference for each measured / computed profile is obtained by appending it with the Spalding’s universal inner layer formulation towards the wallward end up to y+ as low as 0.1. These appended profiles are termed as “true” profiles since they are a better representation of the reality compared to the measured / computed coarseprofiles that completely lack this near-wall velocity field information. Exercises are performed on these true profiles to compute the degradation in M and H with respect to their true values when the data is unavailable below a certain threshold value of y+- FIG. 9A shows Reynolds number variation of the ratio of the value of M to the true value of M, obtained when data points below y+=40 are considered unavailable. FIG. 9B shows a similar plot for H. On the X axis in both the FIGS, is the friction Reynolds number Rcr. In each Fig., the dotted horizontal line marks 1.5% deviation from the ideal value of 1 for the ratio while the dashed horizontal line marks 6.0% deviation. Reynolds number for almost no degradation in M values is Rcr > 500 whereas for H values, it is Rcr > 2000.lt is clear that the degradation in H is much more severe than the degradation in M for a given Reynolds number. The Reynolds number beyond which the degradation becomes negligible, is higher for H (Rer > 2000) than for M (Rer > 500).

[0139] FIG. 10A shows Reynolds number variation of the ratio of the value of M to the true value of M, obtained when data points below y+=80 are considered unavailable. FIG. 10B shows a similar plot for H. On the X axis in both the figures is the friction Reynolds number Rcr. In each figure, the dotted horizontal line marks 1.5% deviation from the ideal value of 1 for the ratio while the dashed horizontal line marks 6.0% deviation. Reynolds number for almost no degradation in M values is Rcr > 2000 whereas for H values, it is Rcr > 8000.

[0140] The study clearly shows that the parameters M and H used in the system and method of the present invention always suffer degradation at lower Reynolds numbers and there is no degradation at high Reynolds numbers. This implies that with increasing Reynolds number, one may afford to lose near-wall data points up to increasingly higher cutoff values of y+ without affecting the true values of M and H, and thus without affecting the accuracy of the system and method of the present invention. Thus the present invention works accurately even at high Reynolds numbers where experimental / computational limitations force the near-wall resolution to be coarse.Referring now to FIG. 11, a flowchart illustrating a method for estimating wall friction in smooth-wall-bounded turbulent flows in accordance with the present invention is described. The method begins with acquisition of flow-related input data including a streamwise mean velocity profile U(y), kinematic viscosity of the fluid, and flow type information (Step SI 102). The flow type is identified (Step SI 103), and in the case of airfoil flows, the streamwise location is validated to ensure that the velocity profile lies within a predefined region of 0.2 < x / c < 0.8 (Step SI 104).

[0141] Upon validation, basic flow parameters including freestream velocity and characteristic length scale are computed from the velocity profile (Step S 1105). For compressible flows, the velocity profile is transformed into an equivalent incompressible profile using a Van Driest transformation prior to further processing (Step SI 106).

[0142] Integral flow parameters comprising kinetic-energy-based parameter M, displacement thickness, momentum thickness, and shape factor H are then computed (Step SI 107). The computed shape factor H is evaluated to determine whether the flow is approaching separation (Step SI 108). Flows satisfying a separation condition are excluded from friction estimation.

[0143] For validated flows not approaching separation, friction velocity is estimated using a semi-empirical friction model based on the computed integral parameters (Step SI 109), and corresponding wall shear stress is determined (Step SI 110). The estimated friction parameters are stored and output for display, analysis, or transmission (Step SI 111), thereby completing the method (Step S1112).

[0144] The present invention therefore provides a technical effect of enabling realtime estimation of wall friction in turbulent flows without requiring near-wall velocity measurements, thereby reducing sensor complexity, computational cost, and experimental constraints in physical flow systems. Advantageously, the present invention provides a system and method for estimating wall friction in smooth-wall turbulent flows. Further, the system and method of the present invention do notrequire near-wall velocity data, especially at high Reynolds numbers where such measurements / computations are formidably challenging, eliminating the use of costly anemometers or photographic techniques. Much faster simulations can be achieved with less computational cost. The system and method of the present invention cater universally to a large variety of flow types such as various kinds of boundary layers, pipe flows, channel flows and even supersonic and hypersonic boundary layers.

[0145] The embodiments were chosen and described in order to best explain the principles of the present invention and its practical application, to thereby enable others, skilled in the art to best utilize the present invention and various embodiments with various modifications as are suited to the particular use contemplated.

[0146] It is understood that various omission and substitutions of equivalents are contemplated as circumstance may suggest or render expedient, but such are intended to cover the application or implementation without departing from the scope of the present invention.

Claims

CLAIMS:

1. A system (100) for estimating wall friction in smooth-wall-bounded turbulent flows estimating wall friction without requiring near-wall velocity measurements, the system comprising:an input unit (104) configured to receive flow-related input data including a streamwise mean velocity profile U(y) and at least one fluid property;a friction estimator (108) connected to the input unit, the friction estimator comprising:a flow check module (204) configured to validate suitability of the received flow-related input data for friction estimation;a parameter module (208) configured to compute flow parameters derived from the velocity profile U(y);a friction engine (216) configured to estimate wall friction using the computed flow parameters while excluding flows approaching separation; anda data module (220) configured to store reference data and computed results;a controller (120) configured to coordinate execution of the flow check module, parameter module, and friction engine; andan output module (136) configured to output estimated wall friction parameters.

2. The system as claimed in claim 1, wherein the flow check module (204) is configured to identify a flow type selected from boundary layer flow, pipe flow, channel flow, or airfoil flow.

3. The system as claimed in claim 1, wherein upon identifying an airfoil flow, the flow check module (204) validates that the velocity profile corresponds to a predefined streamwise region 0.2 < x / c < 0.8 and flags the input data when the validation fails.

4. The system as claimed in claim 1, wherein the parameter module (208) is configured to compute basic flow parameters including freestream orcenterline velocity (Uoo) and a characteristic length scale (5); and integral flow parameters including a kinetic-energy-based parameter (M), displacement thickness (5*), momentum thickness (9), and shape factor (H).

5. The system as claimed in claim 1, wherein the friction engine (216) includes a flow separation check module (217) configured to evaluate the shape factor (H) and exclude flows approaching separation from friction estimation.

6. The system as claimed in claim 1, wherein the friction engine (216) includes a friction compute module (218) configured to estimate friction velocity (UT) and corresponding wall shear stress (TW) using a semi -empirical friction model based on the computed flow parameters.

7. The system as claimed in claim 1 , wherein for compressible turbulent flows, the parameter module (208) is configured to perform a velocity transformation to obtain an equivalent incompressible velocity profile prior to computing the flow parameters.

8. The system as claimed in claim 1, wherein the data module (220) includes a database (221) storing reference parameters and model constants; and a dynamic data module (222) storing received input data, intermediate computed parameters, and estimated wall friction values.

9. The system as claimed in claim 1 , wherein the output module (136) provides at least one of estimated friction velocity and wall shear stress, flow validation status, separation exclusion status, or diagnostic information.

10. A method for estimating wall friction in smooth-wall-bounded turbulent flows without requiring near-wall velocity measurements, the method comprising:receiving flow-related input data including a streamwise mean velocity profile U(y);validating suitability of the received input data for friction estimation;computing flow parameters derived from the velocity profile;excluding flows approaching separation based on the computed parameters;estimating wall friction using a semi-empirical friction model; and outputting estimated wall friction parameters.

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