A double layer papermakers' fabric that is particularly suitable for forming or through-air drying (TAD) of high bulk tissue and towel product is provided. The fabric includes a single warp yarn system interwoven with three weft yarn systems such that: a first of the weft yarn systems is located on the paper side (PS) surface of the fabric; a second of the weft yarn systems is located on the machine side (MS) surface of the fabric; and the third weft yarn system is located intermediate between the first and second weft yarn systems. The yarns of the first and second weft yarn systems are interwoven with the warp yarn system such that they are vertically stacked with respect to one another in the fabric. The yarns of the third weft yarn system are interwoven so as to be located in a central plane of the fabric that is intermediate of the first and second weft yarn systems, and each yarn of the third yarn system is located in between the vertically stacked pairs of weft yarns of the first and second weft yarn systems. The warp and weft yarn systems are interwoven according to an asymmetric design which provides generally rectangular pockets on each of the PS and MS of the fabric with the yarns of the third weft yarn system forming the “bottom” of each pocket. There may be from about 50 to 750 pockets per sq. in. of fabric, each ranging in depth from about 0.1 mm to about 1.0 mm in depth; these pockets impart unevenness to the fabric surface which assists in creating bulk in the sheet formed or conveyed thereon. The fabrics of the invention also offer low sheet contact area, typically less than 30% of the total fabric surface, and generally in the range of from 15% to about 20%. The fabrics are also highly air permeable to ensure good air flow and drainage of the sheet.